A097422 -id:A097422 - cp's OEIS frontend

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 34, 33, 15, 5, 1, 1, 154, 153, 65, 23, 6, 1, 1, 874, 873, 339, 119, 32, 7, 1, 1, 5914, 5913, 2103, 719, 186, 42, 8, 1, 1, 46234, 46233, 15171, 5039, 1230, 267, 54, 9, 1, 1, 409114, 409113, 124755, 40319, 9258, 1891, 380

Offset: 0

Paul D. Hanna, Jan 04 2007

Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, ...;
1,(2), 3, (4), 5, 6, (7), 8, 9, 10, (11), 12, 13, 14, 15, (16), ...;
1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
Columns include:
k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
k=2: A007489 (Sum of k!, k=1..n);
k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
k=4: A033312 (n! - 1);
k=5: Partial sums of A001705;
k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).

Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.

{T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b-1)/2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

A326326 T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 15, 7, 1, 1, 34, 65, 42, 11, 1, 1, 154, 339, 267, 96, 16, 1, 1, 874, 2103, 1891, 831, 191, 22, 1, 1, 5914, 15171, 15023, 7600, 2151, 344, 29, 1, 1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1

Offset: 0

Peter Luschny, Jul 02 2019

			Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 2,     1]
[3] [1, 4,     4,      1]
[4] [1, 10,    15,     7,      1]
[5] [1, 34,    65,     42,     11,    1]
[6] [1, 154,   339,    267,    96,    16,    1]
[7] [1, 874,   2103,   1891,   831,   191,   22,   1]
[8] [1, 5914,  15171,  15023,  7600,  2151,  344,  29,  1]
[9] [1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1]

Same construction for the falling factorial is A176663.
The inverse of the lower triangular matrix is the signed form of A256894.
Second column is A003422(n) and row sums are A003422(n+1).
Alternating row sums are A000007.
Third column is A097422.
Cf. A265609, A132393.

with(PolynomialTools):
T_row := n -> CoefficientList(expand(add(pochhammer(x, j), j=0..n)),x):
ListTools:-Flatten([seq(T_row(n), n=0..9)]);

Table[CoefficientList[FunctionExpand[Sum[Pochhammer[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten

Sum_{k=0..n} T(n, k)*x^k = Sum_{k=0..n} (x)^k, where (x)^k denotes the rising factorial.

Conjecture: T(n,k) = Sum_{i=0..n} A132393(i,k) for 0 <= k <= n. - Werner Schulte, Mar 30 2022

A330016 a(n) = Sum_{k=1..n} (-1)^(n - k) * H(k) * k!, where H(k) is the k-th harmonic number.

Original entry on oeis.org

0, 1, 2, 9, 41, 233, 1531, 11537, 98047, 928529, 9700111, 110843729, 1375599151, 18427159889, 265038487471, 4074124514129, 66660157879471, 1156745432699729, 21220242625821871, 410344904191816529, 8342603132569783471, 177902207647600456529, 3970574571687854263471

Offset: 0

Ilya Gutkovskiy, Nov 27 2019

nonn

Cf. A000254, A001008, A002805, A092692, A097422.

Table[Sum[(-1)^(n - k) HarmonicNumber[k] k!, {k, 1, n}], {n, 0, 22}]

a(n) = Sum_{k=1..n} (-1)^(n - k) * |Stirling1(k+1,2)|.

cp's OEIS Frontend

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

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A326326 T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.

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A330016 a(n) = Sum_{k=1..n} (-1)^(n - k) * H(k) * k!, where H(k) is the k-th harmonic number.

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