A010844 -id:A010844 - cp's OEIS frontend

A056547 a(n) = 6na(n-1) + 1 with a(0)=1.

1, 7, 85, 1531, 36745, 1102351, 39684637, 1666754755, 80004228241, 4320228325015, 259213699500901, 17108104167059467, 1231783500028281625, 96079113002205966751, 8070645492185301207085, 726358094296677108637651

Offset: 0

Henry Bottomley, Jun 20 2000

			a(2) = 6*2*a(1) + 1 = 12*7 + 1 = 85.

Harvey P. Dale, Table of n, a(n) for n = 0..346

Cf. A000522, A010844, A010845, A056545, A056546 for analogs. A056547/(A000142*A000400) is an increasingly good approximation to 6th root of e.

nxt[{n_,a_}]:={n+1,6a(n+1)+1}; NestList[nxt,{0,1},20][[;;,2]] (* Harvey P. Dale, Jul 17 2024 *)

a(n) = floor(e^(1/6)*6^n*n!).
a(n) = n!*Sum_{k=0..n} 6^(n-k)/k!. E.g.f.: exp(x)/(1 - 6*x). - Philippe Deléham, Mar 14 2004
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x = 0..inf} (6*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 6*x) satisfies the differential equation (1 - 6*x)*y' = (7 - 6*x)*y.
a(n) = (6*n + 1)*a(n-1) - 6*(n - 1)*a(n-2).
The sequence b(n) := 6^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = 6^n*n!*( 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/(6*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/6) = 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/((6*n + 1) - ... )))). Cf. A010844. (End)

More terms from James Sellers, Jul 04 2000

A082032 Expansion of e.g.f.: exp(2x)/(1-2x).

Original entry on oeis.org

1, 4, 20, 128, 1040, 10432, 125248, 1753600, 28057856, 505041920, 10100839424, 222218469376, 5333243269120, 138664325005312, 3882601100165120, 116478033004986368, 3727297056159629312, 126728099909427527680, 4562211596739391258624, 173364040676096868352000, 6934561627043874735128576

Offset: 0

Paul Barry, Apr 02 2003

Binomial transform of A010844. a(n) = b such that Integral_{x=0..1} (2*x)^n*exp(-x) dx = c - b*exp(-1). - Francesco Daddi, Jul 31 2011

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000522, A010844, A082033.

With[{nn=30},CoefficientList[Series[Exp[2x]/(1-2x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 02 2021 *)

my(x='x + O('x^25)); Vec(serlaplace(exp(2*x)/(1-2*x))) \\ Michel Marcus, Jan 27 2019

E.g.f.: exp(2*x)/(1-2*x)
a(n) = 2^n*A000522(n). - Vladeta Jovovic, Oct 29 2003
a(n) = 2n*a(n)+2^n, n>0, a(0)=1. - Paul Barry, Aug 26 2004
a(n) +2*(-n-1)*a(n-1) +4*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
G.f.: 1/Q(0), where Q(k)= 1 - 2*x - 2*x*(k+1)/(1-2*x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: 1/Q(0), where Q(k) = 1 - 4*x*(k+1) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = 2^n*hypergeometric_U(1,n+2,1). - Peter Luschny, Nov 26 2014

More terms from Michel Marcus, Jan 27 2019

A263823 a(n) = n!*Sum_{k=0..n} Fibonacci(k-1)/k!, where Fibonacci(-1) = 1, Fibonacci(n) = A000045(n) for n>=0.

Original entry on oeis.org

1, 1, 3, 10, 42, 213, 1283, 8989, 71925, 647346, 6473494, 71208489, 854501957, 11108525585, 155519358423, 2332790376722, 37324646028162, 634518982479741, 11421341684636935, 217005492008104349, 4340109840162091161, 91142306643403921146, 2005130746154886276158

Offset: 0

Vladimir Reshetnikov, Oct 27 2015

nonn

			For n = 3, a(3) = 3!*(Fibonacci(-1)/0! + Fibonacci(0)/1! + Fibonacci(1)/2! + Fibonacci(2)/3!) = 6*(1 + 0 + 1/2 + 1/6) = 10.
For n = 5, Gamma(5+1, phi)*exp(phi) = 120*sqrt(5) + 333 = 240*phi + 213, so a(5) = 213.
G.f. = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 213*x^5 + 1283*x^6 + 8989*x^7 + 71925*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..445
Eric Weisstein's MathWorld, Incomplete Gamma Function.
Eric Weisstein's MathWorld, Fibonacci Number.
Eric Weisstein's MathWorld, Golden Ratio.

Cf. A000045, A111139.

Cf. A009102, A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).

Table[n! Sum[Fibonacci[k-1]/k!, {k, 0, n}], {n, 0, 22}]
Round@Table[(E^(1-GoldenRatio) GoldenRatio Gamma[n+1, 1-GoldenRatio] + E^GoldenRatio Gamma[n+1, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 22}]

a(n) = (Gamma(n+1, 1-phi)*exp(1-phi)*phi+Gamma(n+1, phi)*exp(phi)/phi)/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function, phi=(1+sqrt(5))/2.
a(n) = (phi^(n-1)*hypergeom([1,-n], [], 1-phi)-(-phi)^(1-n)*hypergeom([1,-n], [], phi))/sqrt(5).
Gamma(n+1, phi)*exp(phi) = A111139(n)*phi + a(n).
E.g.f.: (exp(phi*x)/phi+exp(-x/phi)*phi)/(sqrt(5)*(1-x)) = exp(x/2)*(cosh(x*sqrt(5)/2)-sinh(x*sqrt(5)/2)/sqrt(5))/(1-x).
Recurrence: a(0) = 1, a(1) = 1, a(2) = 3, a(n) = (n+1)*a(n-1)+(2-n)*a(n-2)+(2-n)*a(n-3).
a(n) ~ 2*exp(phi-n)*n^(n+1/2)*(1+exp(-sqrt(5))*phi^2)*sqrt(Pi/10)/phi.
0 = a(n)*(+a(n+1) + a(n+2) - 4*a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) + 3*a(n+2) - 5*a(n+3) + a(n+4)) + a(n+2)*(+2*a(n+2) - a(n+4)) + a(n+3)*(+a(n+3)) if n>=0. - Michael Somos, Oct 30 2015

A308876 Expansion of e.g.f. exp(x)(1 - x)/(1 - 2x).

Original entry on oeis.org

1, 2, 7, 40, 317, 3166, 37987, 531812, 8508985, 153161722, 3063234431, 67391157472, 1617387779317, 42052082262230, 1177458303342427, 35323749100272796, 1130359971208729457, 38432239021096801522, 1383560604759484854775, 52575302980860424481432

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Binomial transform of A002866.

Alois P. Heinz, Table of n, a(n) for n = 0..403

Cf. A002866, A010844, A011782, A067273, A161936, A271705, A294466, A327606.

Row sums of A326659.

a:= n-> n! * add(ceil(2^(n-k-1))/k!, k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2019

nmax = 19; CoefficientList[Series[Exp[x] (1 - x)/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[1 + Sum[Binomial[n,k] 2^(k - 1) k!, {k, 1, n}], {n, 0, 19}]

a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 2^(k-1) * k!.
a(n) = A010844(n) - A067273(n).
a(n) ~ n! * 2^(n-1) * exp(1/2). - Vaclav Kotesovec, Jun 29 2019
a(n) = Sum_{k=0..n} k! * A271705(n,k). - Alois P. Heinz, Sep 12 2019

A053482 Binomial transform of A029767.

Original entry on oeis.org

1, 4, 21, 142, 1201, 12336, 149989, 2113546, 33926337, 611660476, 12243073621, 269456124774, 6468249055921, 168191402251432, 4709596238204901, 141291441773619106, 4521383010795364609, 153727989225714801396, 5534225015581836134677

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

This is the column k=3 of an array T(n,k) = A181783(n,k) defined by T(n,0)=T(0,k)=1 and T(n,k) = n*(k-1)*T(n-1,k) +T(n,k-1), which starts
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
   1,   1,   2,   4,   7,  11,  16,  22,  29,  37,  46,...
   1,   1,   5,  21,  63, 151, 311, 575, 981,1573,2401,...
   1,   1,  16, 142, 709,2521,7186,17536,38137,75889,140716,...
   1,   1,  65,1201,9709,50045,193765,614629,1682465,4110913,9176689,...
Column k=2 is A000522. The e.g.f. for column k is E_k(z) = E_(k-1)(z)/[1-(k-1)] = exp(z)/prod_{j=1..k-1} (1-j*z). - Richard Choulet, Dec 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[E^x/(1-3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

E.g.f.: exp(x)*(2/(1-2x)-1/(1-x))=exp(x)/(1-3x+2x^2); a(n)=sum{k=0..n, C(n,k)*k!*(2^(k+1)-1)}; a(n)=n!*sum{k=0..n, (2^(n-k+1)-1)/k!}; a(n)=int(x^n*(exp((1-x)/2)-exp(1-x)),x,1,infty); a(n)=2*A010844(n)-A000522(n); - Paul Barry, Jan 28 2008
Conjecture: a(n) -(3*n+1)*a(n-1) +(2*n+3)*(n-1)*a(n-2) -2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 29 2012
a(n) = 3*n*a(n-2)-2*n*(n-1)*a(n-2)+1, derived from the array defined in the comment, which proves the previous conjecture. - Richard Choulet, Dec 17 2012
a(n) ~ n! * 2^(n+1)*exp(1/2). - Vaclav Kotesovec, Oct 02 2013

A320031 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, 2, 1, 1, 3, 5, 1, 1, 4, 13, 16, 1, 1, 5, 25, 79, 65, 1, 1, 6, 41, 226, 633, 326, 1, 1, 7, 61, 493, 2713, 6331, 1957, 1, 1, 8, 85, 916, 7889, 40696, 75973, 13700, 1, 1, 9, 113, 1531, 18321, 157781, 732529, 1063623, 109601, 1, 1, 10, 145, 2374, 36745, 458026, 3786745, 15383110, 17017969, 986410, 1

Offset: 0

Ilya Gutkovskiy, Oct 03 2018

			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,    2,     3,      4,       5,       6,  ...
  1,    5,    13,     25,      41,      61,  ...
  1,   16,    79,    226,     493,     916,  ...
  1,   65,   633,   2713,    7889,   18321,  ...
  1,  326,  6331,  40696,  157781,  458026,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0..6 give A000012, A000522, A010844, A010845, A056545, A056546, A056547.
Main diagonal gives A277452.
Cf. A007318, A320032, A371898.

A := (n, k) -> simplify(hypergeom([1, -n], [], -k)):
for n from 0 to 5 do seq(A(n, k), k=0..8) od; # Peter Luschny, Oct 03 2018
# second Maple program:
A:= proc(n, k) option remember;
      1 + `if`(n>0, k*n*A(n-1, k), 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 09 2020

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, HypergeometricPFQ[{1, -n}, {}, -k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(x)/(1 - k*x).
A(n,k) = Sum_{j=0..n} binomial(n,j)*j!*k^j.
A(n,k) = hypergeom_2F0([1, -n], [], -k).
A(n,k) = 1 + [n > 0] * k * n * A(n-1,k). - Alois P. Heinz, May 09 2020
A(n,k) = floor(n!*k^n*exp(1/k)), k > 0, n + k > 1. - Peter McNair, Dec 20 2021
From Werner Schulte, Apr 14 2024: (Start)
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371898, i.e., A(n, k) = Sum_{i=0..k} binomial(k, i) * A371898(n, i).
Conjecture: E.g.f. of row n is exp(x) * (Sum_{k=0..n} A371898(n, k) * x^k / k!). (End)

A336804 a(n) = (n!)^2 * Sum_{k=0..n} 2^(n-k) / (k!)^2.

Original entry on oeis.org

1, 3, 25, 451, 14433, 721651, 51958873, 5091969555, 651772103041, 105587080692643, 21117416138528601, 5110414705523921443, 1471799435190889375585, 497468209094520608947731, 195007537965052078707510553, 87753392084273435418379748851, 44929736747147998934210431411713

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A006040, A010844, A228513, A336805, A336807, A336808.

Table[n!^2 Sum[2^(n - k)/k!^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 2*x).

a(0) = 1; a(n) = 2 * n^2 * a(n-1) + 1.

A353546 Expansion of e.g.f. -log(1-2x) exp(x)/2.

Original entry on oeis.org

0, 1, 4, 17, 96, 729, 7060, 83033, 1146656, 18164625, 324488068, 6450956929, 141233271872, 3376008830505, 87480173354964, 2442396780039817, 73089894980585408, 2333809837398044321, 79198287879591647364, 2846319497398561356913

Offset: 0

Seiichi Manyama, May 27 2022

nonn

Cf. A002104, A353547, A353548, A353549.
Cf. A346394.
Essentially partial sums of A010844.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-2*x)*exp(x)/2)))

a(n) = n!*sum(k=0, n-1, 2^(n-1-k)/((n-k)*k!));

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(2*i-1)*v[i]-2*(i-1)*v[i-1]+1); v;

a(n) = n! * Sum_{k=0..n-1} 2^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (2 * n - 1) * a(n-1) - 2 * (n-1) * a(n-2) + 1.
a(n) ~ (n-1)! * exp(1/2) * 2^(n-1). - Vaclav Kotesovec, Jun 08 2022

A007566 a(n+1) = (2n+3)a(n) - 2na(n-1) + 8n, a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 21, 151, 1257, 12651, 151933, 2127231, 34035921, 612646867, 12252937701, 269564629863, 6469551117241, 168208329048891, 4709833213369677, 141294996401091151, 4521439884834917793, 153728956084387206051, 5534242419037939419061, 210301211923441697925687

Offset: 0

N. J. A. Sloane

			1 + 3*x + 21*x^2 + 151*x^3 + 1257*x^4 + 12651*x^5 + 151933*x^6 + 2127231*x^7 + ...

M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 36. [From N. J. A. Sloane, Jan 29 2009]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Doster, Problem 10403, Amer. Math. Monthly, Vol. 101 (1994), p. 792; Solution, Vol. 104 (1997), p. 368.

Cf. A010844.

a:=proc(n) option remember; if n = 0 then RETURN(1); fi; if n = 1 then RETURN(3); fi; (2*n+1)*a(n-1)-(2*n-2)*a(n-2) + 8*(n-1); end;

a(n) = 2*n*a(n-1) + (2*n-1)^2 = 2 * floor(e^(1/2) * n! * 2^n) - (2*n+1) = 2*A010844(n) - (2n+1). - Michael Somos, Mar 26 1999

A056541 a(n) = 2n*a(n-1) + 1 with a(0)=0.

Original entry on oeis.org

0, 1, 5, 31, 249, 2491, 29893, 418503, 6696049, 120528883, 2410577661, 53032708543, 1272785005033, 33092410130859, 926587483664053, 27797624509921591, 889523984317490913, 30243815466794691043

Offset: 0

Henry Bottomley, Jun 20 2000

if s(n) is a sequence defined as s(0)=x, s(n) = 2n*s(n-1)+k, n>0, then s(n) = 2^n*n!*x + a(n)*k. - Gary Detlefs, Feb 20 2010

			a(3) = 2*3*a(2)+1 = 6*5+1 = 31.

Harvey P. Dale, Table of n, a(n) for n = 0..403

Cf. A002627, A173516.

nxt[{n_,a_}]:={n+1,2a(n+1)+1}; NestList[nxt,{0,0},20][[All,2]] (* or *) With[{nn=20},CoefficientList[Series[(Exp[x]-1)/(1-2x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 08 2021 *)

a(n) = floor[(sqrt(e)-1)*2^n*n! ] = A010844(n)-A000165(n).
a(n) = Sum[P(n, k) * 2^k {k=0 to n-1}] - Ross La Haye, Sep 15 2004
Conjecture: a(n) +(-2*n-1)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, May 29 2013
E.g.f.: (exp(x)-1)/(1-2*x) = -12*x/(Q(0)+6*x-3*x^2)/(1-2*x), where Q(k) = 2*(4*k+1)*(32*k^2+16*k+x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013

More terms from James Sellers, Jul 04 2000

cp's OEIS Frontend

A056547 a(n) = 6*n*a(n-1) + 1 with a(0)=1.

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A082032 Expansion of e.g.f.: exp(2*x)/(1-2*x).

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A263823 a(n) = n!*Sum_{k=0..n} Fibonacci(k-1)/k!, where Fibonacci(-1) = 1, Fibonacci(n) = A000045(n) for n>=0.

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A308876 Expansion of e.g.f. exp(x)*(1 - x)/(1 - 2*x).

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A053482 Binomial transform of A029767.

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A320031 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).

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A336804 a(n) = (n!)^2 * Sum_{k=0..n} 2^(n-k) / (k!)^2.

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A353546 Expansion of e.g.f. -log(1-2*x) * exp(x)/2.

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A056547 a(n) = 6na(n-1) + 1 with a(0)=1.

A082032 Expansion of e.g.f.: exp(2x)/(1-2x).

A308876 Expansion of e.g.f. exp(x)(1 - x)/(1 - 2x).

A353546 Expansion of e.g.f. -log(1-2x) exp(x)/2.

A007566 a(n+1) = (2n+3)a(n) - 2na(n-1) + 8n, a(0) = 1, a(1) = 3.