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 1-4 of 4 results.

A357682 a(n) = Sum_{k=0..floor(n/2)} n^k * Stirling2(n,2*k).

Original entry on oeis.org

1, 0, 2, 9, 44, 325, 2742, 24794, 250168, 2796795, 33842610, 439337085, 6100179780, 90139379928, 1409779442190, 23242554452745, 402652762232048, 7308371248274949, 138605556986785674, 2740167375732394378, 56350604098768558140, 1203156656491936711635
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2022

Keywords

Links

Crossrefs

Main diagonal of A357681.
Cf. A242817, A357683.

Programs

  • PARI
    a(n) = sum(k=0, n\2, n^k*stirling(n, 2*k, 2));
  • PARI
    a(n) = round(n!*polcoef(cosh(sqrt(n)*(exp(x+x*O(x^n))-1)), n));
  • PARI
    Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, sqrt(n))+Bell_poly(n, -sqrt(n))))/2;

Formula

a(n) = n! * [x^n] cosh( sqrt(n) * (exp(x) - 1) ).
a(n) = ( Bell_n(sqrt(n)) + Bell_n(-sqrt(n)) )/2, where Bell_n(x) is n-th Bell polynomial.

A357712 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * log(1-x) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 12, 0, 1, 0, 4, 9, 26, 60, 0, 1, 0, 5, 12, 42, 140, 360, 0, 1, 0, 6, 15, 60, 240, 896, 2520, 0, 1, 0, 7, 18, 80, 360, 1614, 6636, 20160, 0, 1, 0, 8, 21, 102, 500, 2520, 12474, 55804, 181440, 0, 1, 0, 9, 24, 126, 660, 3620, 20160, 108900, 525168, 1814400, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  0,   0,   0,   0,   0, ...
  0,  1,   2,   3,   4,   5, ...
  0,  3,   6,   9,  12,  15, ...
  0, 12,  26,  42,  60,  80, ...
  0, 60, 140, 240, 360, 500, ...

Links

Crossrefs

Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.
Main diagonal gives A357683.
Cf. A357681.

Programs

  • PARI
    T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));
  • PARI
    T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;

Formula

T(n,k) = Sum_{j=0..floor(n/2)} k^j * |Stirling1(n,2*j)|.
T(n,k) = ( (sqrt(k))_n + (-sqrt(k))_n )/2, where (x)_n is the Pochhammer symbol.
T(0,k) = 1, T(1,k) = 0; T(n,k) = (2*n-3) * T(n-1,k) - (n^2-4*n+4-k) * T(n-2,k).

A356361 a(n) = Sum_{k=0..floor(n/3)} n^k * |Stirling1(n,3*k)|.

Original entry on oeis.org

1, 0, 0, 3, 24, 175, 1386, 12397, 125664, 1431261, 18099300, 251194911, 3788383248, 61584927495, 1072118178768, 19882255276485, 391068812992512, 8128569896422821, 177984169080865992, 4094103029211918567, 98692513234032009600, 2487731188418039207007
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2022

Keywords

Links

Crossrefs

Cf. A000407, A357683.
Cf. A356362, A356363.
Cf. A079978, A357831.

Programs

  • PARI
    a(n) = sum(k=0, n\3, n^k*abs(stirling(n, 3*k, 1)));
  • PARI
    a(n) = n!*polcoef(sum(k=0, n\3, n^k*(-log(1-x+x*O(x^n)))^(3*k)/(3*k)!), n);
  • PARI
    Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); round(Pochhammer(v, n)+Pochhammer(v*w, n)+Pochhammer(v*w^2, n))/3;

Formula

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . a(n) = n! * [x^n] F(-n^(1/3) * log(1-x)).
a(n) = ( (n^(1/3))_n + (n^(1/3)*w)_n + (n^(1/3)*w^2)_n )/3, where (x)_n is the Pochhammer symbol.

A357721 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling1(n,2*k).

Original entry on oeis.org

1, 0, -2, 9, -28, 0, 1200, -16464, 167904, -1393200, 7429240, 43124400, -2404571904, 55590286752, -1027511503200, 16489054310400, -222885864448000, 1994839594780032, 14489184835474272, -1470395490046560000, 54581408106475622400, -1608207353670788640000
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2022

Keywords

Links

Crossrefs

Main diagonal of A357720.
Cf. A357683, A357729.

Programs

  • PARI
    a(n) = sum(k=0, n\2, (-n)^k*stirling(n, 2*k, 1));
  • PARI
    a(n) = round(n!*polcoef(cos(sqrt(n)*log(1+x+x*O(x^n))), n));
  • PARI
    a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(n)*I+k)+prod(k=0, n-1, -sqrt(n)*I+k)))/2;

Formula

a(n) = n! * [x^n] cos( sqrt(n) * log(1+x) ).
a(n) = (-1)^n * ( (sqrt(n) * i)_n + (-sqrt(n) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
Showing 1-4 of 4 results.

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