A084948 -id:A084948 - cp's OEIS frontend

A034908 One half of octo-factorial numbers.

1, 10, 180, 4680, 159120, 6683040, 334152000, 19380816000, 1279133856000, 94655905344000, 7761784238208000, 698560581438720000, 68458936980994560000, 7256647319985423360000, 827257794478338263040000, 100925450926357268090880000, 13120308620426444851814400000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..333
Index entries for sequences related to factorial numbers.

Cf. A007696, A045755, A034909, A039410, A039411, A039412, A084948.

[(&*[(8*k-6): k in [1..n]])/2: n in [1..30]]; // G. C. Greubel, Feb 26 2018

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 8*x)^(-1/4))/2, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 26 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1+(1-8*x)^(-1/4))/2)) \\ G. C. Greubel, Feb 26 2018

2*a(n) = (8*n-6)(!^8) := product(8*j-6, j=1..n) = 2^n*A007696(n); compare with A007696(n) = (4*n-3)(!^4) := product(4*j-3, j=1..n).
E.g.f.: (-1+(1-8*x)^(-1/4))/2.
G.f.: x/(1-10x/(1-8x/(1-18x/(1-16x/(1-26x/(1-24x/(1-34x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A084948(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). (End)

Terms a(16) onward added by G. C. Greubel, Feb 26 2018

A167884 Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 8.

Original entry on oeis.org

1, 1, 1, 1, 18, 1, 1, 179, 179, 1, 1, 1636, 6086, 1636, 1, 1, 14757, 144362, 144362, 14757, 1, 1, 132854, 2941135, 7218100, 2941135, 132854, 1, 1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1, 1, 10761672, 1001178268, 9211047544, 18315657030, 9211047544, 1001178268, 10761672, 1

Offset: 1

Roger L. Bagula, Nov 14 2009

			Triangle begins as:
  1;
  1,       1;
  1,      18,        1;
  1,     179,      179,         1;
  1,    1636,     6086,      1636,         1;
  1,   14757,   144362,    144362,     14757,        1;
  1,  132854,  2941135,   7218100,   2941135,   132854,       1;
  1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_8(n,k)

For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A167884, ...

Cf. A084948 (row sums).

T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
A167884[n_, k_]:= T[n,k,8];
Table[A167884[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)

@CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A167884(n,k): return T(n,k,8)
flatten([[ A167884(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 18 2022

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 8.

Sum_{k=1..n} T(n, k) = A084948(n-1).

Edited by N. J. A. Sloane, May 08 2013

cp's OEIS Frontend

A034908 One half of octo-factorial numbers.

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A167884 Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 8.

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cp's OEIS Frontend

A034908 One half of octo-factorial numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A167884 Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A167884 Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = 8.