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.

A320032 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(-x)/(1 - k*x).

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 5, 2, 1, 1, 3, 13, 29, 9, -1, 1, 4, 25, 116, 233, 44, 1, 1, 5, 41, 299, 1393, 2329, 265, -1, 1, 6, 61, 614, 4785, 20894, 27949, 1854, 1, 1, 7, 85, 1097, 12281, 95699, 376093, 391285, 14833, -1, 1, 8, 113, 1784, 26329, 307024, 2296777, 7897952, 6260561, 133496, 1
Offset: 0

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Author

Ilya Gutkovskiy, Oct 03 2018

Keywords

Comments

For n > 0 and k > 0, A(n,k) gives the number of derangements of the generalized symmetric group S(k,n), which is the wreath product of Z_k by S_n. - Peter Kagey, Apr 07 2020

Examples

			E.g.f. of column k: A_k(x) = 1 + (k - 1)*x/1! + (2*k^2 - 2*k + 1)*x^2/2! + (6*k^3 - 6*k^2 + 3*k - 1)*x^3/3! + (24*k^4 - 24*k^3 + 12*k^2 - 4*k + 1)*x^4/4! + ...
Square array begins:
   1,   1,     1,      1,      1,       1,  ...
  -1,   0,     1,      2,      3,       4,  ...
   1,   1,     5,     13,     25,      41,  ...
  -1,   2,    29,    116,    299,     614,  ...
   1,   9,   233,   1393,   4785,   12281,  ...
  -1,  44,  2329,  20894,  95699,  307024,  ...

Links

Crossrefs

Columns k=0..5 give A033999, A000166, A000354, A000180, A001907, A001908.
Main diagonal gives A319392.
Cf. A320031.

Programs

  • Maple
    A:= proc(n, k) option remember;
     `if`(n=0, 1, k*n*A(n-1, k)+(-1)^n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 07 2020
  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[-x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, (-1)^n HypergeometricPFQ[{1, -n}, {}, k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: exp(-x)/(1 - k*x).
A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j!*k^j.
A(n,k) = (-1)^n*2F0(1,-n; ; k).

A277452 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * k!.

Original entry on oeis.org

1, 2, 13, 226, 7889, 458026, 39684637, 4788052298, 766526598721, 157108817646514, 40104442275129101, 12472587843118746322, 4641978487740740993233, 2036813028164774540010266, 1040451608604560812273060189, 612098707457003526384666111226
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 16 2016

Keywords

Links

Crossrefs

Main diagonal of A320031.
Cf. A086331, A277373, A277423, A277453.

Programs

  • Maple
    a := n -> simplify(hypergeom([1, -n], [], -n)):
seq(a(n), n=0..15); # Peter Luschny, Oct 03 2018
# second Maple program:
b:= proc(n, k) option remember;
      1 + `if`(n>0, k*n*b(n-1, k), 0)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 09 2020
  • Mathematica
    Flatten[{1, Table[Sum[Binomial[n, k]*n^k*k!, {k, 0, n}], {n, 1, 20}]}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * n^k * k!); \\ Michel Marcus, Sep 18 2018

Formula

a(n) = exp(1/n) * n^n * Gamma(n+1, 1/n).
a(n) ~ n^n * n!.
a(n) = n! * [x^n] exp(x)/(1 - n*x). - Ilya Gutkovskiy, Sep 18 2018
a(n) = floor(n^n*n!*exp(1/n)), n > 0. - Peter McNair, Dec 20 2021

A371898 Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 15, 48, 36, 1, 64, 504, 1008, 576, 1, 325, 5680, 22680, 31680, 14400, 1, 1956, 72060, 510480, 1304640, 1382400, 518400, 1, 13699, 1036224, 12233340, 50823360, 94046400, 79833600, 25401600, 1, 109600, 16798768, 318469536, 2017814400, 5794790400, 8346240000, 5893171200, 1625702400
Offset: 0

Views

Author

Werner Schulte, Apr 11 2024

Keywords

Examples

			Lower triangular array starts:
n\k :  0      1        2         3         4         5         6         7
==========================================================================
  0 :  1
  1 :  1      1
  2 :  1      4        4
  3 :  1     15       48        36
  4 :  1     64      504      1008       576
  5 :  1    325     5680     22680     31680     14400
  6 :  1   1956    72060    510480   1304640   1382400    518400
  7 :  1  13699  1036224  12233340  50823360  94046400  79833600  25401600
  etc.

Crossrefs

Cf. A000012 (column 0), A007526 (column 1), A001044 (main diagonal).
Cf. A000166, A048993, A131689, A320031, A371766, A371767.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*HypergeometricPFQ[{1, -n}, {}, -j], {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten  (* Peter Luschny, Apr 12 2024 *)
  • PARI
    T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))

Formula

T(n, k) = Sum_{i=k..n} A131689(i, k) * n! / (n-i)!.
T(n, k) = n! * k! * (Sum_{i=0..n-k} A048993(n-i, k) / i!).
T(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A320031(n, i).
Conjecture: E.g.f. of column k is exp(t) * t^k * k! / (Prod_{i=0..k} (1 - i*t)).
Conjecture: Sum_{k=0..n} (-1)^(n-k) * T(n, k) = A000166(n).
T(n, k) = A371766(n, k) * A371767(n, k). - Peter Luschny, Apr 14 2024
Showing 1-3 of 3 results.

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