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 1-7 of 7 results.

A130191 Square of the Stirling2 matrix A048993.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 15, 32, 12, 1, 0, 52, 175, 110, 20, 1, 0, 203, 1012, 945, 280, 30, 1, 0, 877, 6230, 8092, 3465, 595, 42, 1, 0, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 0, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1
Offset: 0

Views

Author

Wolfdieter Lang, Jun 01 2007

Keywords

Comments

Without row n=0 and column k=0 this is triangle A039810.
This is an associated Sheffer matrix with e.g.f. of the m-th column ((exp(f(x))-1)^m)/m! with f(x)=:exp(x)-1.
The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - Peter Luschny, Apr 19 2015
Also the Bell transform of shifted Bell numbers A000110(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

Examples

			Triangle starts:
  1;
  0,   1;
  0,   2,    1;
  0,   5,    6,    1;
  0,  15,   32,   12,    1;
  0,  52,  175,  110,   20,   1;
  0, 203, 1012,  945,  280,  30,  1;
  0, 877, 6230, 8092, 3465, 595, 42, 1;

Links

Crossrefs

Columns k=0..3 give A000007, A000110 (for n > 0), A000558, A000559.
Row sums: A000258.
Alternating row sums: A130410.
T(2n,n) gives A321712.
Cf. A039810 (another version), A048993.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> combinat:-bell(n+1), 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[BellB[# + 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
a[n_, m_]:= Sum[StirlingS2[n, k]*StirlingS2[k, m], {k,m,n}]; Table[a[n, m], {n, 0, 100}, {m, 0, n}]//Flatten (* G. C. Greubel, Jul 10 2018 *)
  • PARI
    for(n=0, 9, for(k=0, n, print1(sum(j=k, n, stirling(n, j, 2)*stirling(j, k, 2)), ", "))) \\ G. C. Greubel, Jul 10 2018
  • Sage
    # uses[riordan_array from A256893]
riordan_array(1, exp(exp(x) - 1), 8, exp=true) # Peter Luschny, Apr 19 2015

Formula

a(n,k) = Sum_{j=k..n} S2(n,j) * S2(j,k), n>=k>=0.
E.g.f. row polynomials with argument x: exp(x*f(f(z))).
E.g.f. column k: ((exp(exp(x) - 1) - 1)^k)/k!.

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!.

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!.

A099391 Expansion of e.g.f. 1/(2 - exp(exp(exp(x) - 1) - 1)).

Original entry on oeis.org

1, 1, 5, 36, 342, 4048, 57437, 950512, 17975438, 382424397, 9039989107, 235062317196, 6667866337309, 204905200542916, 6781157167505291, 240446179599065951, 9094120016963808935, 365453749501228063845
Offset: 0

Views

Author

Ralf Stephan, Oct 18 2004

Keywords

Links

Crossrefs

Column k=3 of A363007.
Row p=3 of A153278 (for n>=1).
Cf. A000110, A083355, A130410.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(2-Exp[Exp[Exp[x]-1]-1]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 10 2014 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(exp(exp(x)-1)-1)))) \\ Seiichi Manyama, May 12 2023

Formula

(1/2) Sum[k=0..inf, k^n/k! * Sum[r=1..inf, e^(-r)r^k/r!*Li(-r, 1/2e) ]], with Li the polylogarithm.
a(n) ~ n! / (2 * (1 + log(2)) * (1 + log(1 + log(2))) * log(1 + log(1 + log(2)))^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(n) = Sum_{k=0..n} Stirling2(n,k) * A083355(k). - Seiichi Manyama, May 12 2023

Extensions

Definition clarified by Harvey P. Dale, Apr 10 2014

A308518 Expansion of e.g.f. exp(1 - exp(1 - exp(x))).

Original entry on oeis.org

1, 1, 1, 0, -4, -12, -3, 150, 744, 525, -16799, -118280, -148289, 4036802, 37244157, 68676153, -1758280309, -20207442595, -49855713746, 1245931950070, 17250366460410, 53991885230741, -1330935478357842, -21705274324058996, -83339285813776419, 2026672671500822591, 38327819123289163864
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 03 2019

Keywords

Crossrefs

Cf. A000258, A000587, A130410, A308519.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[1 - Exp[1 - Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] BellB[k, -1], {k, 0, n}], {n, 0, 26}]
a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] BellB[k, -1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]

Formula

a(n) = Sum_{k=0..n} (-1)^k*Stirling2(n,k)*A000587(k).
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1)*A000587(k)*a(n-k).

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 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.