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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A277406 a(n) equals the sum of all permutations of compositions of functions (1 + k*x) for k=1..n, evaluated at x=1.

Original entry on oeis.org

1, 2, 9, 76, 1100, 25176, 846132, 39321696, 2413753344, 189030205440, 18383301319680, 2172771551093760, 306662748175330560, 50933260598106862080, 9832247390118248121600, 2182733403365330313523200, 552134185815355910465126400, 157863713952139655599757721600, 50654908373638564216229105664000
Offset: 0

Views

Author

Paul D. Hanna, Oct 16 2016

Keywords

Examples

			Illustration of initial terms.
a(0) = 1, by convention;
a(1) = 2, the function (1+x) evaluated at x=1;
a(2) = 9, the sum of permutations of compositions of functions (1+x) and (1+2*x), evaluated at x=1: (1+x)o(1+2*x) + (1+2*x)o(1+x) = (2*x + 2) + (2*x + 3) = 4*x + 5.
a(3) = 76, the sum of permutations of compositions of functions (1+x), (1+2*x), and (1+3*x), evaluated at x=1: (1+x)o(1+2*x)o(1+3*x) + (1+x)o(1+3*x)o(1+2*x) + (1+2*x)o(1+1*x)o(1+3*x) + (1+2*x)o(1+3*x)o(1+1*x) + (1+3*x)o(1+1*x)o(1+2*x) + (1+3*x)o(1+2*x)o(1+1*x) = (6*x + 4) + (6*x + 5) + (6*x + 5) + (6*x + 9) + (6*x + 7) + (6*x + 10) = 36*x + 40.
etc.
Alternatively,
a(1) = 2 = Sum_{i=1..1} (1+i),
a(2) = 9 = Sum_{i=1..2, j=1..2, j<>i} (1 + i*(1+j)),
a(3) = 76 = Sum_{i=1..3, j=1..3, k=1..3, i,j,k distinct} (1 + i*(1 + j*(1+k))),
a(4) = 1100 = Sum_{i=1..4, j=1..4, k=1..4, m=1..4, i,j,k,m distinct} (1 + i*(1 + j*(1 + k*(1+m)))), etc.

Crossrefs

Cf. A277405, A277407, A277408.

Programs

  • Mathematica
    Table[Sum[k!*(n-k)! * Sum[(-1)^(n-i+1) * StirlingS2[i, n-k+1] * StirlingS1[n+1, i], {i, 0, n-k+1}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 27 2016 *)
  • PARI
    {a(n) = sum(k=0,n, k!*(n-k)! * sum(i=0, n-k+1, (-1)^(n-i+1) * stirling(i, n-k+1, 2) * stirling(n+1, i, 1)))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} k!*(n-k)! * Sum_{i=0..n-k+1} (-1)^(n-i+1) * Stirling2(i,n-k+1) * Stirling1(n+1,i).
a(n) = (n!)^2 + A277405(n).
a(n) = (n+1) * A277407(n).
a(n) = Sum_{k=0..n} A277408(n,k).

A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 11, 12, 6, 24, 50, 70, 60, 24, 120, 274, 450, 510, 360, 120, 720, 1764, 3248, 4410, 4200, 2520, 720, 5040, 13068, 26264, 40614, 47040, 38640, 20160, 5040, 40320, 109584, 236248, 403704, 538776, 544320, 393120, 181440, 40320
Offset: 1

Views

Author

N. J. A. Sloane, Apr 14 2011

Keywords

Comments

Also the coefficients of the polynomials which are generated by the exponential generating function -log(1 + x*log(1 - t)). The polynomials might be called 'logarithmic polynomials'. Note also A003713, and A263634 for a different use of this term. See the paper of F. Qi for a related, but different family of polynomials. - Peter Luschny, Jul 11 2020
Edgar remarks that these coefficients are related to Stirling numbers of the second kind (cf. A008277).
The first column and the main diagonal are the factorials (A000142). The n-th entry on the first subdiagonal is A001710(n+1). The second column is A000254, the third column is 2*A000399, and the fourth column is 6*A000454. In general, the k-th column is (k-1)!*s(n,k), where s(n,k) is the unsigned Stirling number of the first kind. - Nathaniel Johnston, Apr 15 2011
With offset n=0, k=0 : triangle T(n,k), read by rows,given by T(n,k) = k*T(n-1, k-1) + n*T(n-1, k) with T(0, 0) = 1. - Philippe Deléham, Oct 04 2011

Examples

			Triangle begins:
1
1    1
2    3    2
6    11   12   6
24   50   70   60   24
120  274  450  510  360  120
...

Links

Crossrefs

Cf. A277408, A003713, A263634.

Programs

  • Maple
    S:=proc(n,k)global s:if(n=0 and k=0)then s[0,0]:=1:elif(n=0 or k=0)then s[n,k]:=0:elif(not type(s[n,k],integer))then s[n,k]:=(n-1)*S(n-1,k)+S(n-1,k-1):fi:return s[n,k]:end:
T:=proc(n,k)return (k-1)!*S(n,k);end:
for n from 1 to 6 do for k from 1 to n do print(T(n,k)):od:od: # Nathaniel Johnston, Apr 15 2011
# With offset n = 0, k = 0:
A188881 := (n, k) -> k!*abs(Stirling1(n+1, k+1)):
seq(seq(A188881(n,k), k=0..n), n=0..8); # Peter Luschny, Oct 19 2017
# Alternative:
gf := -log(1 + x*log(1 - t)): ser := series(gf, t, 18):
toeff := n -> n!*expand(coeff(ser, t, n)):
seq(print(seq(coeff(toeff(n), x, k), k=1..n)), n=1..8); # Peter Luschny, Jul 10 2020
  • Mathematica
    Table[(k - 1)! * Sum[StirlingS2[i, k] * (-1)^(n - i) * StirlingS1[n, i], {i, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Apr 17 2015 *)
  • Maxima
    T(n,k):=(k-1)!*sum(stirling2(i,k)*(-1)^(n-i)*stirling1(n,i),i,0,k); /* Vladimir Kruchinin, Apr 17 2015 */
  • PARI
    {T(n, k) = if( k<1 || k>n, 0, (n-1)! * polcoeff( (x / (1 - exp(-x * (1 + x * O(x^n)))))^n, n-k))}; /* Michael Somos, May 10 2017 */
  • PARI
    {T(n, k) = if( k<1 || n<0, 0, (k-1)! * sum(i=0, k, stirling(i, k, 2) * (-1)^(n-i) * stirling(n, i, 1)))}; /* Michael Somos, May 10 2017 */

Formula

T(n, k) = (k-1)!*Sum_{i=0..k}(Stirling2(i,k)*(-1)^(n-i)*Stirling1(n,i)) =
T(n, k) = Sum_{i=0..k}(W(i,k)*(-1)^(n-i)*Stirling1(n,i)), where W(n,k) is the Worpitzky triangle A028246. - Vladimir Kruchinin, Apr 17 2015.
T(n,k) = [x^k] n!*[t^n](-log(1 + x*log(1 - t))). - Peter Luschny, Jul 10 2020
T(n,k) = Sum_{m=0..n-k} abs(Stirling1(n-1,m+k-1))*(k+m-1)!/m!. - Vladimir Kruchinin, Jul 14 2025

Extensions

a(11)-a(45) from Nathaniel Johnston, Apr 15 2011
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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