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 1-5 of 5 results.

A343832 a(n) = Sum_{k=0..n} k! * binomial(n,k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 4, 31, 358, 5509, 106096, 2456299, 66471826, 2059640713, 71920704124, 2794938616471, 119653108240414, 5595650767265101, 283841520215780008, 15523069639558351459, 910529206043204428426, 57023540590242398853649, 3797750659849704886903156, 268025698704886063968108943
Offset: 0

Views

Author

Seiichi Manyama, May 01 2021

Keywords

Comments

Let A(x) be the e.g.f. of this sequence, and B(x) be the e.g.f. of A082545, then A(x)/B(x) = C(x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108). This follows from the fact that this sequence and A082545 form adjacent semi-diagonals of table A088699. - Paul D. Hanna, Aug 16 2022

Links

Crossrefs

Cf. A000522, A045721, A082545, A152059, A248668, A343830, A343831, A000108.

Programs

  • Magma
    [Factorial(n)*Evaluate(LaguerrePolynomial(n, n+1), -1): n in [0..40]]; // G. C. Greubel, Aug 11 2022
  • Maple
    a := n -> add(k!*binomial(n, k)*binomial(2*n+1, k), k=0..n):
a := n -> n!*add(binomial(2*n+1, k)/(n-k)!, k=0..n):
a := n -> (-1)^n*KummerU(-n, n+2, -1):
a := n -> n!*LaguerreL(n, n+1, -1): # Peter Luschny, May 02 2021
  • Mathematica
    a[n_] := Sum[k! * Binomial[n, k] * Binomial[2*n+1, k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, May 01 2021 *)
Table[(-1)^n * HypergeometricU[-n, 2 + n, -1], {n, 0, 20}] (* Vaclav Kotesovec, May 02 2021 *)
  • PARI
    a(n) = sum(k=0, n, k!*binomial(n, k)*binomial(2*n+1, k));
  • PARI
    a(n) = (2*n+1)!*sum(k=0, n, binomial(n, k)/(k+n+1)!);
  • PARI
    a(n) = n!*sum(k=0, n, binomial(2*n+1, k)/(n-k)!);
  • PARI
    a(n) = n!*pollaguerre(n, n+1, -1);
  • SageMath
    [factorial(n)*gen_laguerre(n, n+1, -1) for n in (0..40)] # G. C. Greubel, Aug 11 2022

Formula

a(n) = (2*n+1)! * Sum_{k=0..n} binomial(n,k)/(k+n+1)!.
a(n) = n! * Sum_{k=0..n} binomial(2*n+1,k)/(n-k)!.
a(n) = n! * LaguerreL(n, n+1, -1).
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+2).
a(n) == 1 (mod 3).
a(n) ~ 2^(2*n + 3/2) * n^n / exp(n-1). - Vaclav Kotesovec, May 02 2021
From Paul D. Hanna, Aug 16 2022: (Start)
E.g.f.: exp( (1-2*x - sqrt(1-4*x))/(2*x) ) * (1 - sqrt(1-4*x)) / (2*x*sqrt(1-4*x)), derived from the e.g.f for A082545 given by Mark van Hoeij.
E.g.f.: exp(C(x) - 1) * C(x) / sqrt(1-4*x), where C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). (End)

A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 13, 34, 73, 1, 5, 21, 73, 209, 501, 1, 6, 31, 136, 501, 1546, 4051, 1, 7, 43, 229, 1045, 4051, 13327, 37633, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 394353, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 4596553
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2017

Keywords

Examples

			Square array begins:
    1,    1,    1,    1,     1, ... A000012;
    1,    2,    3,    4,     5, ... A000027;
    3,    7,   13,   21,    31, ... A002061;
   13,   34,   73,  136,   229, ... A135859;
   73,  209,  501, 1045,  1961, ...
  501, 1546, 4051, 9276, 19081, ...
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 13;
  1, 4, 13, 34,  73;
  1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021

Links

Crossrefs

Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.
Main diagonal gives A152059.
Similar table: A086885, A088699, A176120.

Programs

  • Magma
    function t(n,k)
  if n eq 0 then return 1;
  else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);
  end if; return t;
end function;
[t(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021
  • Mathematica
    t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];
T[n_,k_]:= t[k,n-k]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)
  • Sage
    @CachedFunction
def t(n,k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j,k)/factorial(n-j) for j in (1..n) )
def T(n,k): return t(k,n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

Formula

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.
A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.
From Seiichi Manyama, Jan 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} binomial(n+k-1,j)/(n-j)!.
A(n,k) = n! * LaguerreL(n, k-1, -1). (End)

A380491 a(n) = n! * Sum_{k=0..n} binomial(2*n-3,k)/(n-k)!.

Original entry on oeis.org

1, 0, 3, 34, 501, 9276, 207775, 5470158, 165625929, 5671386136, 216730118331, 9144481575450, 422249317829053, 21180324426577044, 1146880568461500951, 66677192513929212166, 4142571510546929867025, 273910161452560881843888, 19204878684852222745880179
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2025

Keywords

Links

Crossrefs

Cf. A082545, A152059, A251568, A293985, A343832, A380492, A380493.

Programs

  • PARI
    a(n) = n!*pollaguerre(n, n-3, -1);

Formula

a(n) = n! * LaguerreL(n, n-3, -1).
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n-2).

A380492 a(n) = n! * Sum_{k=0..n} binomial(2*n-2,k)/(n-k)!.

Original entry on oeis.org

1, 1, 7, 73, 1045, 19081, 424051, 11109337, 335262313, 11453449105, 436944953791, 18412283563081, 849345673881277, 42570185481576793, 2303643608370636715, 133859418832759525081, 8312945340897388101841, 549460711493172343519777, 38513032385247860120975863
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2025

Keywords

Links

Crossrefs

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380493.

Programs

  • PARI
    a(n) = n!*pollaguerre(n, n-2, -1);

Formula

a(n) = n! * LaguerreL(n, n-2, -1).
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n-1).

A380493 a(n) = n! * Sum_{k=0..n} binomial(2*n+3,k)/(n-k)!.

Original entry on oeis.org

1, 6, 57, 748, 12585, 259026, 6315001, 178134552, 5711078673, 205209960670, 8171229107481, 357235056697476, 17014791129640057, 877089297426429738, 48657292133825026905, 2890717184573264397616, 183125115830192864360481, 12323226433255671469949622
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2025

Keywords

Links

Crossrefs

Cf. A082545, A152059, A251568, A293985, A343832, A380491, A380492.

Programs

  • PARI
    a(n) = n!*pollaguerre(n, n+3, -1);

Formula

a(n) = n! * LaguerreL(n, n+3, -1).
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+4).
Showing 1-5 of 5 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.