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.

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A089900 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the factorials, starting with row 0: {1!,2!,3!,...}.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 11, 4, 1, 120, 49, 18, 5, 1, 720, 261, 92, 27, 6, 1, 5040, 1631, 536, 159, 38, 7, 1, 40320, 11743, 3552, 1029, 256, 51, 8, 1, 362880, 95901, 26608, 7353, 1848, 389, 66, 9, 1, 3628800, 876809, 223456, 58095, 14384, 3125, 564, 83, 10, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2003

Keywords

Comments

Row 1 is A001339, antidiagonal sums form A089902 and the main diagonal is A089901; the next lower diagonal forms {1,4,27,256,..,n^n,..}, which is the hyperbinomial transform (cf. A088956) of the main diagonal.

Examples

			Note secondary diagonal: {(n+1)^(n+1)}; rows begin:
1, 2,. 6,. 24,. 120,.. 720,.. 5040,..
1, 3, 11,. 49,. 261,. 1631,. 11743,..
1,_4, 18,. 92,. 536,. 3552,. 26608,..
1, 5,_27, 159, 1029,. 7353,. 58095,..
1, 6, 38,_256, 1848, 14384, 121264,..
1, 7, 51, 389,_3125, 26595, 241015,..
1, 8, 66, 564, 5016,_46656, 456048,..
1, 9, 83, 787, 7701, 78077,_823543,..

Crossrefs

Cf. A001339, A088956, A089901, A089902.

Programs

  • Mathematica
    t[n_, k_] := (n^(k+2) - Exp[n]*(n-k-1)*Gamma[k+2, n])/(k+1) // Round; Table[t[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jun 24 2013 *)
  • PARI
    T(n,k)=if(n<0 || k<0,0,sum(i=0,k,n^(k-i)*binomial(k,i)*(i+1)!))

Formula

T(0, k)=(k+1)!, T(n+1, n)=(n+1)^(n+1), T(n, k)=sum_{i=0..k}n^(k-i)*binomial(k, i)*(i+1)!
E.g.f.: 1/((1-y*exp(x))*(1-x)^2). E.g.f. (n-th row): exp(n*x)/(1-x)^2.

A089901 Main diagonal of A089900, also the inverse hyperbinomial of A000312 (offset 1).

Original entry on oeis.org

1, 3, 18, 159, 1848, 26595, 456048, 9073911, 205437312, 5214027267, 146602156800, 4522866752943, 151895344131072, 5516066815430691, 215373243256915968, 8996883483108522375, 400372897193449586688, 18908951043963993686019
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2003

Keywords

Comments

The n-th row of array A089900 is the n-th binomial transform of the factorials found in row 0: {1!,2!,3!,...,(n+1)!,...}. The hyperbinomial transform of this main diagonal gives: {1,4,27,...,(n+1)^(n+1),...}, which is the next lower diagonal in array A089900.
a(n), for n>=1, is the number of colored labeled mappings from n points to themselves, where each component is one of three colors. - Steven Finch, Nov 28 2021

Links

Crossrefs

Cf. A000312, A089900, A089902, A000312.

Programs

  • Mathematica
    CoefficientList[Series[1/(1+LambertW[-x])^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
Flatten[{1,Table[Sum[n^(n-k)*Binomial[n,k]*(k+1)!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 09 2013 *)
a[n_] := (n^(n + 2) + Exp[n] Gamma[n + 2, n]) / (n + 1);
Table[a[n], {n, 0, 17}]  (* Peter Luschny, Nov 29 2021 *)
  • PARI
    /* As (n+1)-th term of the n-th binomial transform of {(n+1)!}: */
{a(n)=if(n<0,0,sum(i=0,n,n^(n-i)*binomial(n,i)*(i+1)!))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* As (n+1)-th term of inverse hyperbinomial of {(n+1)^(n+1)}: */
{a(n)=if(n<0,0,sum(i=0,n,-(n-i-1)^(n-i-1)*binomial(n,i)*(i+1)^(i+1)))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} n^(n-k) * C(n,k) * (k+1)!.
a(n) = Sum_{k=0..n} -(n-k-1)^(n-k-1) * C(n,k) * (k+1)^(k+1).
E.g.f.: 1 / (1 + LambertW(-x))^3.
E.g.f.: (Sum_{n>=0} (n+1)^(n+1) * x^n/n!) * (Sum_{n>=0} -(n-1)^(n-1) * x^n/n!).
a(n) ~ n^(n+1) * (1 + sqrt(Pi/(2*n))). - Vaclav Kotesovec, Jul 09 2013
a(n) = (n^(n + 2) + exp(n)*Gamma(n + 2, n)) / (n + 1). - Peter Luschny, Nov 29 2021
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