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 1-6 of 6 results.

A351429 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, 1, -6, 1, -1, 0, -1, 24, 1, -1, -1, 1, 1, -120, 1, -1, -2, 0, 1, -1, 720, 1, -1, -3, -4, 6, -2, 1, -5040, 1, -1, -4, -11, -2, 32, -9, -1, 40320, 1, -1, -5, -21, -41, 76, 115, -9, 1, -362880, 1, -1, -6, -34, -129, -75, 953, 172, 50, -1, 3628800
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Examples

			Square array begins:
     1,  1,  1,   1,   1,    1,     1, ...
    -1, -1, -1,  -1,  -1,   -1,    -1, ...
     2,  1,  0,  -1,  -2,   -3,    -4, ...
    -6, -1,  1,   0,  -4,  -11,   -21, ...
    24,  1,  1,   6,  -2,  -41,  -129, ...
  -120, -1, -2,  32,  76,  -75,  -806, ...
   720,  1, -9, 115, 953, 1540, -3334, ...

Crossrefs

Columns k=0..5 give A133942, A033999, A000587, A130410, A351427, A351428.
Main diagonal gives A351433.
Cf. A111933, A351420.

Programs

  • Maple
    A:= (n, k)-> n!*(g->coeff(series(1/(1+(g@@k)(x)), x, n+1), x, n))(x->exp(x)-1):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # 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}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* 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)));

Formula

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.

A351424 a(n) = n! * [x^n] -log(1 - f^(n-1)(x)), where f(x) = log(1+x).

Original entry on oeis.org

1, 0, 3, -48, 1270, -50375, 2803829, -208616562, 20003317746, -2402323535658, 353219463307920, -62411008199372327, 13048469028962425266, -3186116313706825820802, 898478811755719496052919, -289795933163271680910773018, 106008143082108931457543700504
Offset: 1

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Links

Crossrefs

Main diagonal of A351420.
Cf. A139383, A302358, A351433.

Programs

  • Maple
    g:= x-> log(1+x):
a:= n-> n! * coeff(series(-log(1-(g@@(n-1))(x)), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022
  • Mathematica
    T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; a[n_] := T[n, n]; Array[a, 16] (* 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)));
a(n) = T(n, n);

Formula

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

A351422 Expansion of e.g.f. -log(1 - log(1 + log(1 + log(1+x)))).

Original entry on oeis.org

1, -2, 8, -48, 386, -3905, 47701, -683592, 11250291, -209168071, 4336482905, -99197868847, 2481962140797, -67426166949102, 1976463051528507, -62178381389729317, 2089532143617395264, -74702625442877063902, 2830904065389397804534, -113348477836878447492630
Offset: 1

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Crossrefs

Column k=4 of A351420.
Cf. A000310, A351427.

Programs

  • Mathematica
    T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; a[n_] := T[n, 4]; Array[a, 20] (* Amiram Eldar, Feb 11 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(-log(1-log(1+log(1+log(1+x))))))
  • PARI
    T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));
a(n) = T(n, 4);

Formula

a(n) = T(n,4), T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.

A351423 Expansion of e.g.f. -log(1 - log(1 + log(1 + log(1 + log(1+x))))).

Original entry on oeis.org

1, -3, 16, -124, 1270, -16243, 249776, -4494334, 92716855, -2158505443, 55996266046, -1602132913687, 50124833578256, -1702501170925098, 62391472267252322, -2453892459756494459, 103101294099324376489, -4608723131704380915202
Offset: 1

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Crossrefs

Column k=5 of A351420.
Cf. A000359, A351428.

Programs

  • Mathematica
    T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; a[n_] := T[n, 5]; Array[a, 18] (* Amiram Eldar, Feb 11 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1-log(1+log(1+log(1+log(1+x)))))))
  • PARI
    T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));
a(n) = T(n, 5);

Formula

a(n) = T(n,5), T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.

A351421 Expansion of e.g.f. -log(1 - log(1 + log(1+x))).

Original entry on oeis.org

1, -1, 3, -13, 77, -576, 5219, -55567, 680028, -9405302, 145067040, -2468571128, 45936991110, -927915150852, 20219040931738, -472697857817078, 11801903989774760, -313395752536945568, 8819464678850030936, -262185434197432956664, 8210080944919085511680
Offset: 1

Views

Author

Seiichi Manyama, Feb 11 2022

Keywords

Crossrefs

Column k=3 of A351420.
Cf. A000268, A130410.

Programs

  • Mathematica
    T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; a[n_] := T[n, 3]; Array[a, 21] (* Amiram Eldar, Feb 11 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(-log(1-log(1+log(1+x)))))
  • PARI
    T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));
a(n) = T(n, 3);

Formula

a(n) = T(n,3), T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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