A095000 -id:A095000 - cp's OEIS frontend

A001341 Expansion of e.g.f. 6*exp(x)/(1-x)^4.

6, 30, 174, 1158, 8742, 74046, 696750, 7219974, 81762438, 1005151902, 13336264686, 189992451270, 2893180308774, 46904155833918, 806663460996462, 14669947577257926, 281298999630211590, 5672559830998316574, 120014233288249367598, 2658221288671765756422

Offset: 0

N. J. A. Sloane and Simon Plouffe

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35.
Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages.

Cf. A095000.

nn = 20; Range[0, nn]! CoefficientList[Series[6*Exp[x]/(1 - x)^4, {x, 0, nn}], x] (* T. D. Noe, Jun 28 2012 *)

my(x='x+O('x^66)); Vec(serlaplace(6*exp(x)/(1-x)^4)) \\ Joerg Arndt, May 09 2013

a(n) = ceiling( n!*(n^3 + 3*n^2 + 5*n + 2)*exp(1) ). - Mark van Hoeij, Nov 11 2009
G.f.: Q(0)*(1-x)^2/x^3 - 2/x + 1/x^2 - 1/x^3, where Q(k)= 1 + (2*k + 1)*x/( 1 - x - 2*x*(1-x)*(k+1)/(2*x*(k+1) + (1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013
a(n) = 6 * A095000(n). - Alois P. Heinz, Jan 17 2025

Error in definition corrected Jan 30 2008

More terms from N. J. A. Sloane, Jan 30 2008

A095722 E.g.f.: exp(x)/(1-x)^8.

Original entry on oeis.org

1, 9, 89, 961, 11265, 142601, 1940089, 28245729, 438351041, 7226001865, 126122874201, 2324074591169, 45094140207169, 919088049256521, 19633713260950265, 438708172312264801, 10234490436580101249

Offset: 0

Philippe Deléham, Jul 08 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n) for x = 1, 2, 3, 4, 5, 6, 7 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..250

With[{nn=20},CoefficientList[Series[Exp[x]/(1-x)^8,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 26 2013 *)
Table[HypergeometricPFQ[{8, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*8^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+7)! / 7!.
a(n) = 2F0(8,-n;;-1). - Benedict W. J. Irwin, May 27 2016

A095740 E.g.f.: exp(x)/(1-x)^9.

Original entry on oeis.org

1, 10, 109, 1288, 16417, 224686, 3288205, 51263164, 848456353, 14862109042, 274743964621, 5346258202000, 109249238631169, 2339328151461718, 52384307381414317, 1224472783033479556, 29826054965115774145

Offset: 0

Philippe Deléham Jul 09 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n), A095722(n) for x = 1, 2, 3, 4, 5, 6, 7, 8.

Robert Israel, Table of n, a(n) for n = 0..440

seq(simplify(hypergeom([9,-n],[],-1)),n=0..30); # Robert Israel, May 27 2016

Table[HypergeometricPFQ[{9, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*9^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+8)!/8!.
a(n) = 2F0(9,-n;;-1). - Benedict W. J. Irwin, May 27 2016
a(n) = ((n^8 + 28*n^7 + 350*n^6 + 2492*n^5 + 10899*n^4 + 29596*n^3 + 48082*n^2 + 42048*n + 14833) * Gamma(n+1,1)*e + n^7 + 28*n^6 + 349*n^5 + 2465*n^4 + 10579*n^3 + 27501*n^2 + 40132*n + 25487) / 40320. - Robert Israel, May 27 2016

cp's OEIS Frontend

A001341 Expansion of e.g.f. 6*exp(x)/(1-x)^4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A095722 E.g.f.: exp(x)/(1-x)^8.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A095740 E.g.f.: exp(x)/(1-x)^9.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Formula