cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A351420 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - f^(k-1)(x)), where f(x) = log(1+x).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 1, 6, 1, -2, 3, -1, 24, 1, -3, 8, -13, 8, 120, 1, -4, 16, -48, 77, -26, 720, 1, -5, 27, -124, 386, -576, 194, 5040, 1, -6, 41, -259, 1270, -3905, 5219, -1142, 40320, 1, -7, 58, -471, 3244, -16243, 47701, -55567, 9736, 362880
Offset: 1

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Examples

			Square array begins:
    1,   1,    1,     1,      1,      1, ...
    1,   0,   -1,    -2,     -3,     -4, ...
    2,   1,    3,     8,     16,     27, ...
    6,  -1,  -13,   -48,   -124,   -259, ...
   24,   8,   77,   386,   1270,   3244, ...
  120, -26, -576, -3905, -16243, -50375, ...

Crossrefs

Columns k=1..5 give A000142(n-1), (-1)^(n-1) * A089064(n), A351421, A351422, A351423.
Main diagonal gives A351424.
Cf. A111933, A351429.

Programs

  • Mathematica
    T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Feb 11 2022 *)
  • PARI
    T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));

Formula

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

A351427 Expansion of e.g.f. 1/exp(exp(exp(exp(x)-1)-1)-1).

Original entry on oeis.org

1, -1, -2, -4, -2, 76, 953, 9103, 77054, 550457, 2123247, -32551171, -1197444063, -26019611323, -478608682879, -7915791047153, -115777452314939, -1320533985179144, -3550854626237496, 455708391448493954, 21276221692251262984, 703173682906460544467
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Crossrefs

Column k=4 of A351429.
Cf. A000587, A130410, A351428.
Cf. A000307, A351422.

Programs

  • Mathematica
    T[n_, 0] := (-1)^n * n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; a[n_] := T[n, 4]; Array[a, 22, 0] (* Amiram Eldar, Feb 11 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/exp(exp(exp(exp(x)-1)-1)-1)))
  • PARI
    T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = T(n, 4);

Formula

a(n) = T(n,4), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = (-1)^n * n!.

A351428 Expansion of e.g.f. 1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1).

Original entry on oeis.org

1, -1, -3, -11, -41, -75, 1540, 37725, 657715, 10551750, 163089430, 2407275470, 31865298262, 290682880132, -2479867505029, -267542605513289, -11438897571729494, -404343336811199242, -13192591498632627584, -410340915410006575406, -12233989907129223814578
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Crossrefs

Column k=5 of A351429.
Cf. A000587, A130410, A351427.
Cf. A000357, A351423.

Programs

  • Maple
    g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1/((g@@5)(x)+1), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 11 2022
  • Mathematica
    T[n_, 0] := (-1)^n * n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; a[n_] := T[n, 5]; Array[a, 20, 0] (* Amiram Eldar, Feb 11 2022 *)
With[{nn=20},CoefficientList[Series[1/Exp[Exp[Exp[Exp[Exp[x]-1]-1]-1]-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 09 2025 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1)))
  • PARI
    T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = T(n, 5);

Formula

a(n) = T(n,5), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = (-1)^n * n!.

A351433 a(n) = n! * [x^n] 1/(1 + f^n(x)), where f(x) = exp(x) - 1.

Original entry on oeis.org

1, -1, 0, 0, -2, -75, -3334, -192864, -14443260, -1372372623, -162009663365, -23314158802286, -4022712394579207, -820399656345934444, -195326656416326556562, -53709209673236813446542, -16896296201917398543629108, -6030879950631118091070849321
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Links

Crossrefs

Main diagonal of A351429.
Cf. A139383, A302358, A351424.

Programs

  • Maple
    g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1/(1+(g@@n)(x)), x, n+1), x, n):
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 11 2022
  • Mathematica
    T[n_, 0] := (-1)^n * n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; a[n_] := T[n, n]; Array[a, 17, 0] (* Amiram Eldar, Feb 11 2022 *)
  • PARI
    T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = T(n, n);

Formula

a(n) = T(n,n), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = (-1)^n * n!.

A363007 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - f^k(x)), where f(x) = exp(x) - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 24, 1, 1, 5, 23, 75, 120, 1, 1, 6, 36, 175, 541, 720, 1, 1, 7, 52, 342, 1662, 4683, 5040, 1, 1, 8, 71, 594, 4048, 18937, 47293, 40320, 1, 1, 9, 93, 949, 8444, 57437, 251729, 545835, 362880, 1, 1, 10, 118, 1425, 15775, 143783, 950512, 3824282, 7087261, 3628800
Offset: 0

Views

Author

Seiichi Manyama, May 12 2023

Keywords

Examples

			Square array begins:
    1,   1,    1,    1,    1,     1, ...
    1,   1,    1,    1,    1,     1, ...
    2,   3,    4,    5,    6,     7, ...
    6,  13,   23,   36,   52,    71, ...
   24,  75,  175,  342,  594,   949, ...
  120, 541, 1662, 4048, 8444, 15775, ...

Crossrefs

Columns k=0..5 give A000142, A000670, A083355, A099391, A363008, A363009.
Main diagonal gives A363010.
Cf. A153278, A351420, A351429.

Programs

  • PARI
    T(n, k) = if(k==0, n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));

Formula

T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.
T(n,k) = A153278(k,n) for n >= 1 and k >= 1.
Showing 1-5 of 5 results.

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