All series from Math Book/Series Enclopedia "Iksan"(1868)

Sequence	Definition	Closed form	First terms
Triangular numbers	T(n) = \sum_{k=1}^{n} k	\binom{n+1}{2}	1, 3, 6, 10, 15, \dots
Tetrahedral numbers	\sum_{k=1}^{n} T(k)	\binom{n+2}{3}	1, 4, 10, 20, 35, \dots
Pentatope (4-simplex) numbers	\sum_{k=1}^{n} \binom{k+2}{3}	\binom{n+3}{4}	1, 5, 15, 35, 70, \dots
5-simplex numbers	\sum_{k=1}^{n} \binom{k+3}{4}	\binom{n+4}{5}	1, 6, 21, 56, 126, \dots
Simplex numbers (general)	S(n, d) = \binom{n + d}{d + 1}	\binom{n+d}{d+1}	d=2:\ 1, 4, 10, 20, 35, \dots
Square pyramidal numbers	\sum_{k=1}^{n} k^2	\dfrac{n(n+1)(2n+1)}{6}	1, 5, 14, 30, 55, \dots
Sums of square pyramidal numbers	\sum_{k=1}^{n} \dfrac{k(k+1)(2k+1)}{6}	\dfrac{n(n+1)^2(n+2)}{12}	1, 6, 20, 50, 105, \dots
Second-iterated square pyramidal sums	\sum_{k=1}^{n} \sum_{j=1}^{k} \dfrac{j(j+1)(2j+1)}{6}	\dfrac{n(n+1)(n+2)(n+3)(2n+3)}{120}	1, 7, 27, 77, 182, \dots
Centered polygonal numbers	1 + s \cdot T(n)	1 + \dfrac{s\, n(n+1)}{2}	s=5:\ 1, 6, 16, 31, 51, \dots

Weighted and iterated sums

Sequence	Definition	Closed form	First terms
Index-weighted triangular sums	\sum_{k=1}^{n} k \cdot T(k)	\dfrac{n(n+1)(n+2)(3n+1)}{24}	1, 7, 25, 65, 115, \dots
Index-weighted tetrahedral sums	\sum_{k=1}^{n} k \cdot \binom{k+2}{3}	\dfrac{n(n+1)(n+2)(n+3)(4n+1)}{120}	1, 9, 39, 119, 294, \dots
Index-weighted pyramidal sums	\sum_{k=1}^{n} k \cdot \dfrac{k(k+1)(2k+1)}{6}	\dfrac{n(n+1)(n+2)(8n^2+11n+1)}{120}	1, 11, 53, 173, 448, \dots
Sums of cubes (Nicomedes)	\sum_{k=1}^{n} k^3	\left(\dfrac{n(n+1)}{2}\right)^2	1, 9, 36, 100, 225, \dots
Double sums of cubes	\sum_{i=1}^{n} \sum_{k=1}^{i} k^3	\dfrac{n(n+1)(n+2)(3n^2+6n+1)}{60}	1, 10, 46, 146, 371, \dots
Power sums (general)	\sum_{k=1}^{n} k^p	\sum_{j} j! \, S(p, j) \binom{n+1}{j+1}	p=2:\ 1, 5, 14, 30, 55, \dots
Iterated power sums	\underbrace{\sum \cdots \sum}_{\text{depth}} k^p	iterated Stirling basis	p=3,\ \text{depth}=2:\ 1, 10, 46, 146, 371, \dots
Iterated square sums (general)	\underbrace{\sum \cdots \sum}_{d} k^2	\binom{n+d}{d+1} \cdot \dfrac{2n+d}{d+2}	d=1:\ 1, 5, 14, 30, 55, \dots

Pile and frustum sequences

Sequence	Definition	Closed form	First terms
Rectangular frustum sums	\sum_{i=0}^{m-1} (a+i)(b+i)	discrete prismoidal formula	a=b=1:\ 1, 5, 14, 30, 55, \dots
Arithmetic series sums	\sum_{i=0}^{m-1} (a + i\,d)	\dfrac{m\,(2a + (m-1)d)}{2}	a=1, d=2:\ 1, 4, 9, 16, 25, \dots
Cone-pile layer counts	A_k: centered hexagonal \leftrightarrow triple squares	layer terms of \left\lceil \frac{n}{2} \right\rceil^3 + \left\lfloor \frac{n}{2} \right\rfloor \left( \left\lfloor \frac{n}{2} \right\rfloor + 1 \right) \left( 2 \left\lfloor \frac{n}{2} \right\rfloor + 1 \right) / 2	1, 3, 7, 12, 19, 27, 37, \dots
Cone-pile layer sums	\sum_{k=1}^{n} A_k	\dfrac{(2n+1)\binom{n+1}{2} + \lceil n/2 \rceil}{4}	1, 4, 11, 23, 42, 69, \dots
Double accumulation of cone piles	\sum_{i=1}^{n} W_i, W_i = \sum_{k=1}^{i} A_k	N^2(N+1)^2 + \frac{N(N+1)}{2} if n = 2N; \ N^4 if n = 2N - 1	1, 5, 16, 39, 81, \dots
Truncated pile sums	\sum_{k=1}^{n} k \sum_{j=1}^{m} (m + j - 1)^2	simplex-basis finite-difference expansion	—
Generalized truncated pile sums	\sum_{k=1}^{n} k \cdot P(m + k - 1), P(j) = \sum_{i=1}^{j} i^2	2A_2 - A_1 in ascending-weighted simplex basis	—

Combinatorial building blocks

Sequence	Definition	Closed form	First terms
Binomial coefficients	C(n, k)	\dfrac{n!}{k!\,(n-k)!}	n=5 row: 1, 5, 10, 10, 5, 1
Multiset coefficients	M(n, k) = \binom{n + k - 1}{k}	\binom{n+k-1}{k}	n=4:\ 1, 4, 10, 20, 35, \dots
Stirling numbers of the second kind	partitions of n elements into k non-empty subsets	\dfrac{1}{k!} \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} j^n	k=2:\ 1, 3, 7, 15, 31, \dots

Notation: T(n) = \binom{n+1}{2}, S(n, k) = \binom{n+k}{k+1}, S(p, j) Stirling numbers of the second kind, P(j) square pyramidal numbers.