A308878 -id:A308878 - cp's OEIS frontend

A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)log(1 + x))/(1 - klog(1 + x)).

1, 1, 1, 1, 1, -1, 1, 1, 1, 2, 1, 1, 3, 2, -6, 1, 1, 5, 14, 4, 24, 1, 1, 7, 38, 86, 14, -120, 1, 1, 9, 74, 384, 664, 38, 720, 1, 1, 11, 122, 1042, 4854, 6136, 216, -5040, 1, 1, 13, 182, 2204, 18344, 73614, 66240, 600, 40320, 1, 1, 15, 254, 4014, 49774, 387512, 1302552, 816672, 6240, -362880

Offset: 0

Seiichi Manyama, Jun 12 2020

			Square array begins:
   1,  1,   1,    1,     1,     1, ...
   1,  1,   1,    1,     1,     1, ...
  -1,  1,   3,    5,     7,     9, ...
   2,  2,  14,   38,    74,   122, ...
  -6,  4,  86,  384,  1042,  2204, ...
  24, 14, 664, 4854, 18344, 49774, ...

Columns k=1..3 give A006252, A308878, A335530.
Main diagonal gives A335529.
Cf. A320080.

T[0, k_] = 1; T[n_, k_] := Sum[If[k == 0 && j <= 1, 1, k^(j - 1)] * j! * StirlingS1[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

T(0,k)=1 and T(n,k) = Sum_{j=0..n} j! * k^(j-1) * Stirling1(n,j) for n > 0.

A308877 Expansion of e.g.f. (1 + log(1 - x))/(1 + 2*log(1 - x)).

Original entry on oeis.org

1, 1, 5, 38, 386, 4904, 74776, 1330272, 27046848, 618653280, 15723024864, 439559609664, 13405656582336, 442915145716224, 15759326934391296, 600783539885546496, 24430204949876794368, 1055516761826050203648, 48286612866726631489536, 2331682676308057000255488

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Cf. A001710, A002866, A008275, A011782, A050351, A088500, A320349, A308878.

nmax = 19; CoefficientList[Series[(1 + Log[1 - x])/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] 2^(k - 1) k!, {k, 1, n}], {n, 1, 19}]]

a(0) = 1; a(n) = Sum_{k=1..n} |Stirling1(n,k)| * 2^(k-1) * k!.

a(n) ~ n! * exp(n/2) / (4 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Jun 29 2019

cp's OEIS Frontend

A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)log(1 + x))/(1 - klog(1 + x)).

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A308877 Expansion of e.g.f. (1 + log(1 - x))/(1 + 2*log(1 - x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)*log(1 + x))/(1 - k*log(1 + x)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A308877 Expansion of e.g.f. (1 + log(1 - x))/(1 + 2*log(1 - x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A334369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - (k-1)log(1 + x))/(1 - klog(1 + x)).