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 1-3 of 3 results.

A059370 Triangle of numbers obtained by inverting infinite matrix defined in A059369, read from right to left.

Original entry on oeis.org

1, 1, -2, 1, -4, 2, 1, -6, 8, -4, 1, -8, 18, -16, -4, 1, -10, 32, -44, 12, -48, 1, -12, 50, -96, 72, -96, -336, 1, -14, 72, -180, 216, -216, -480, -2928, 1, -16, 98, -304, 500, -544, -376, -4672, -28144, 1, -18, 128, -476, 996, -1312, 256, -5856, -45520, -298528
Offset: 0

Views

Author

N. J. A. Sloane, Jan 28 2001

Keywords

Examples

			Triangle starts
1;
1,  -2;
1,  -4,   2;
1,  -6,   8,   -4;
1,  -8,  18,  -16,  -4;
1, -10,  32,  -44,  12,   -48;
1, -12,  50,  -96,  72,   -96, -336;
1, -14,  72, -180, 216,  -216, -480, -2928;
1, -16,  98, -304, 500,  -544, -376, -4672, -28144;
1, -18, 128, -476, 996, -1312,  256, -5856, -45520, -298528;
... - _Joerg Arndt_, Apr 20 2013

Crossrefs

Cf. A059369, A059372, A059373.

Programs

  • Mathematica
    nmax = 10; t[n_, k_] := t[n, k] = Sum[(m+1)!*t[n-m-1, k-1], {m, 0, n-k}]; t[n_, 1] = n!; t[n_, n_] = 1; tnk = Table[t[n, k], {n, 1, nmax}, {k, 1, nmax}]; Reverse /@ Inverse[tnk] // DeleteCases[#, 0, 2]& // Flatten (* Jean-François Alcover, Jun 14 2013 *)

Extensions

More terms from Vladeta Jovovic, Mar 05 2001

A090238 Triangle T(n, k) read by rows. T(n, k) is the number of lists of k unlabeled permutations whose total length is n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 16, 6, 1, 0, 120, 72, 30, 8, 1, 0, 720, 372, 152, 48, 10, 1, 0, 5040, 2208, 828, 272, 70, 12, 1, 0, 40320, 14976, 4968, 1576, 440, 96, 14, 1, 0, 362880, 115200, 33192, 9696, 2720, 664, 126, 16, 1, 0, 3628800, 996480, 247968, 64704, 17312, 4380, 952, 160, 18, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 23 2004, Jun 14 2007

Keywords

Comments

T(n,k) is the number of lists of k unlabeled permutations whose total length is n. Unlabeled means each permutation is on an initial segment of the positive integers. Example: with dashes separating permutations, T(3,2) = 4 counts 1-12, 1-21, 12-1, 21-1. - David Callan, Nov 29 2007
For n > 0, -Sum_{i=0..n} (-1)^i*T(n,i) is the number of indecomposable permutations A003319. - Peter Luschny, Mar 13 2009
Also the convolution triangle of the factorial numbers for n >= 1. - Peter Luschny, Oct 09 2022

Examples

			Triangle begins:
  1;
  0,       1;
  0,       2,      1;
  0,       6,      4,      1;
  0,      24,     16,      6,     1;
  0,     120,     72,     30,     8,     1;
  0,     720,    372,    152,    48,    10,     1;
  0,    5040,   2208,    828,   272,    70,    12,    1;
  0,   40320,  14976,   4968,  1576,   440,    96,   14,   1;
  0,  366880, 115200,  33192,  9696,  2720,   664,  126,  16,   1;
  0, 3628800, 996480, 247968, 64704, 17312,  4380,  952, 160,  18,  1;
  ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Crossrefs

Another version: A059369.
Row sums: A051296, A003319 (n>0).
Diagonals: A000007, A000142, A059371, A000012, A005843, A054000.
Cf. A084938.

Programs

  • Maple
    T := proc(n,k) option remember; if n=0 and k=0 then return 1 fi;
if n>0 and k=0 or k>0 and n=0 then return 0 fi;
T(n-1,k-1)+(n+k-1)*T(n-1,k)/k end:
for n from 0 to 10 do seq(T(n,k),k=0..n) od; # Peter Luschny, Mar 03 2016
# Uses function PMatrix from A357368.
PMatrix(10, factorial); # Peter Luschny, Oct 09 2022
  • Mathematica
    T[n_, k_] := T[n, k] = T[n-1, k-1] + ((n+k-1)/k)*T[n-1, k]; T[0, 0] = 1; T[, 0] = T[0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018 *)

Formula

T(n, k) is given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
T(n, k) = T(n-1, k-1) + ((n+k-1)/k)*T(n-1, k); T(0, 0)=1, T(n, 0)=0 if n > 0, T(0, k)=0 if k > 0.
G.f. for the k-th column: (Sum_{i>=1} i!*t^i)^k = Sum_{n>=k} T(n, k)*t^n.
Sum_{k=0..n} T(n, k)*binomial(m, k) = A084938(m+n, m). - Philippe Deléham, Jan 31 2004
T(n, k) = Sum_{j>=0} A090753(j)*T(n-1, k+j-1). - Philippe Deléham, Feb 18 2004
From Peter Bala, May 27 2017: (Start)
Conjectural o.g.f.: 1/(1 + t - t*F(x)) = 1 + t*x + (2*t + t^2)*x^2 + (6*t + 4*t^2 + t^3)*x^3 + ..., where F(x) = Sum_{n >= 0} n!*x^n.
If true then a continued fraction representation of the o.g.f. is 1 - t + t/(1 - x/(1 - t*x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End)

Extensions

New name using a comment from David Callan by Peter Luschny, Sep 01 2022

A059371 a(n) = (n-1)! + ((n+1)/2)*a(n-1), a(1)=0.

Original entry on oeis.org

1, 4, 16, 72, 372, 2208, 14976, 115200, 996480, 9607680, 102366720, 1195568640, 15193785600, 208728576000, 3081867264000, 48659595264000, 817953583104000, 14581909536768000, 274755150544896000, 5455208664170496000, 113825841809670144000
Offset: 2

Views

Author

N. J. A. Sloane, Jan 28 2001

Keywords

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Links

Crossrefs

Second diagonal of triangle in A059369.

Programs

  • Maple
    series(hypergeom([1,2],[],x)^2, x=0, 30);  # Mark van Hoeij, Apr 20 2013
  • Mathematica
    Rest[Rest[CoefficientList[Series[(x^2-2*x-2*Log[1-x])/(x-2)^2, {x, 0, 20}], x]* Range[0, 20]!]] (* Vaclav Kotesovec, Aug 11 2013 *)
Table[-2 n! - 2 (n + 1)! Re[LerchPhi[2, 1, 2 + n]], {n, 2, 10}] (* Vladimir Reshetnikov, Oct 17 2015 *)
Table[2*Sum[(2^k - 1) * Abs[StirlingS1[n, k]] * BernoulliB[k], {k, 0, n}], {n, 3,
25}] (* Vaclav Kotesovec, Oct 04 2022 *)
  • PARI
    a(n)=sum(i=1,n-1,i!*(n-i)!) \\ Jon Perry, May 06 2006
  • PARI
    { a=0; for (n = 2, 200, write("b059371.txt", n, " ", a = (n - 1)! + a*(n + 1)/2); ) } \\ Harry J. Smith, Jun 26 2009

Formula

E.g.f.: (x^2-2*x-2*log(1-x))/(x-2)^2. - Vladeta Jovovic, May 04 2003
a(n) = Sum_{i=1..n-1} i!*(n-i)!. E.g., a(6) = 1!*5!+2!4!+3!3!+4!2!+5!1! = 120+48+36+48+120 = 372. - Jon Perry, May 06 2006
a(n) = 2*Integral_{t>=0}t^n*exp(-t)*(t*exp(-t)*Ei(t)-1), with Ei the exponential integral function.
Recurrence: 2*a(n) = (3*n-1)*a(n-1) - (n-1)*n*a(n-2). - Vaclav Kotesovec, Aug 11 2013
a(n) ~ 2*(n-1)!. - Vaclav Kotesovec, Aug 11 2013
a(n) = -2*n! - 2*(n+1)!*Re(LerchPhi(2, 1, 2 + n)). - Vladimir Reshetnikov, Oct 17 2015
a(n) = n!*Re(hypergeom([1,1],[n+2],2) - 1). - Vladimir Reshetnikov, Oct 19 2015

Extensions

Better description from Vladeta Jovovic, May 04 2003
More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001
Showing 1-3 of 3 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.