A056545 -id:A056545 - cp's OEIS frontend

A010845 a(n) = 3na(n-1) + 1, a(0) = 1.

1, 4, 25, 226, 2713, 40696, 732529, 15383110, 369194641, 9968255308, 299047659241, 9868572754954, 355268619178345, 13855476147955456, 581929998214129153, 26186849919635811886, 1256968796142518970529

Offset: 0

Simon Plouffe

a(n)/(A000142*A000244) is an increasingly good approximation to cube root of e.
Related to Incomplete Gamma Function at 1/3. - Michael Somos, Mar 26 1999
For positive n, a(n) equals 3^n times the permanent of the n X n matrix with (4/3)'s along the main diagonal, and 1's everywhere else. - John M. Campbell, Jul 10 2011

			1 + 4*x + 25*x^2 + 226*x^3 + 2713*x^4 + 40696*x^5 + 732529*x^6 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Cf. A000522, A010844, A056545, A056546, A056547 for analogs.

Table[ Gamma[ n, 1/3 ]*Exp[ 1/3 ]*3^(n-1), {n, 1, 24} ]
a[ n_] := If[ n<0, 0, Floor[ n! E^(1/3) 3^n ]] (* Michael Somos, Sep 04 2013 *)
Range[0, 20]! CoefficientList[Series[Exp[x]/(1 - 3 x), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 17 2014 *)

{a(n) = if( n<0, 0, n! * sum(k=0, n, 3^(n-k) / k!))} /* Michael Somos, Sep 04 2013 */

E.g.f.: exp(x)/(1-3*x).
a(n) = floor( n!*e^(1/3)*3^n ) = n! * (Sum_{k=0..n} 3^(n-k) / k!) = n! * (e^(1/3) * 3^n - Sum_{k>n} 3^(n-k) / k!). - Michael Somos, Mar 26 1999
a(n) = Sum_{k=0..n} P(n, k)*3^k. - Ross La Haye, Aug 29 2005
Binomial transform of A032031. - Carl Najafi, Sep 11 2011
Conjecture: a(n) +(-3*n-1)*a(n-1) +3*(n-1)*a(n-2)=0. - R. J. Mathar, Feb 16 2014
a(n) = hypergeometric_U(1,n+2,1/3)/3. - Peter Luschny, Nov 26 2014
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x >= 0} (3*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 3*x) satisfies the differential equation (1 - 3*x)*y' = (4 - 3*x)*y. Mathar's recurrence above follows easily from this.
The sequence b(n) := (3^n)*n! also satisfies Mathar's recurrence with b(0) = 1, b(1) = 3. This leads to the continued fraction representation a(n) = (3^n)*n!*( 1 + 1/(3 - 3/(7 - 6/(10 - ... - (3*n - 3)/(3*n + 1) )))) for n >= 2. Taking the limit as n -> oo gives the continued fraction representation exp(1/3) = 1 + 1/(3 - 3/(7 - 6/(10 - ... - (3*n - 3)/((3*n + 1) - ... )))). Cf. A010844. (End)

Better description and formulas from Michael Somos

More terms from James Sellers, Jul 04 2000

A056546 a(n) = 5na(n-1) + 1 with a(0)=1.

Original entry on oeis.org

1, 6, 61, 916, 18321, 458026, 13740781, 480927336, 19237093441, 865669204846, 43283460242301, 2380590313326556, 142835418799593361, 9284302221973568466, 649901155538149792621, 48742586665361234446576

Offset: 0

Henry Bottomley, Jun 20 2000

			a(2) = 5*2*a(1) + 1 = 10*6 + 1 = 61.

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, A010845, A056545, A056547 for analogs. A056546/(A000142*A000351) is an increasingly good approximation to 5th root of e.

m = 16; CoefficientList[E^x/(1-5x) + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Jun 03 2019 *)

a(n) = floor(e^(1/5)*5^n*n!).
From Philippe Deléham, Mar 14 2004: (Start)
a(n) = n!*Sum_{k=0..n} 5^(n-k)/k!.
E.g.f.: exp(x)/(1 - 5*x). (End)
a(n) = Sum_{k=0..n} P(n, k)*5^k. - Ross La Haye, Aug 29 2005
a(n) = hypergeometric_U(1, n+2 , 1/5)/5. - Peter Luschny, Nov 26 2014
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x >= 0} (5*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 5*x) satisfies the differential equation (1 - 5*x)*y' = (6 - 5*x)*y.
a(n) = (5*n + 1)*a(n-1) - 5*(n - 1)*a(n-2).
The sequence b(n) := 5^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 5. This leads to the continued fraction representation a(n) = 5^n*n!*( 1 + 1/(5 - 5/(11 - 10/(16 - ... - (5*n - 5)/(5*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/5) = 1 + 1/(5 - 5/(11 - 10/(16 - ... - (5*n - 5)/((5*n + 1) - ... )))). Cf. A010844. (End)

More terms from James Sellers, Jul 04 2000

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

Original entry on oeis.org

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

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)

A336807 a(n) = (n!)^2 * Sum_{k=0..n} 4^(n-k) / (k!)^2.

Original entry on oeis.org

1, 5, 81, 2917, 186689, 18668901, 2688321745, 526911062021, 134889231877377, 43704111128270149, 17481644451308059601, 8461115914433100846885, 4873602766713466087805761, 3294555470298303075356694437, 2582931488713869611079648438609, 2324638339842482649971683594748101

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A006040, A056545, A336804, A336805, A336808.

Table[n!^2 Sum[4^(n - k)/k!^2, {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 4 x), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 4*x).

a(0) = 1; a(n) = 4 * n^2 * a(n-1) + 1.

A353548 Expansion of e.g.f. -log(1-4x) exp(x)/4.

Original entry on oeis.org

0, 1, 6, 47, 540, 8429, 166210, 3952955, 109981816, 3502905369, 125648153278, 5011458069639, 219987094389524, 10538817637744005, 547118005892177018, 30595552548140425747, 1833501625083035349488, 117219490267316310468913

Offset: 0

Seiichi Manyama, May 27 2022

nonn

Cf. A002104, A353546, A353547, A353549.
Cf. A346396.
Essentially partial sums of A056545.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-4*x)*exp(x)/4)))

a(n) = n!*sum(k=0, n-1, 4^(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]=(4*i-3)*v[i]-4*(i-1)*v[i-1]+1); v;

a(n) = n! * Sum_{k=0..n-1} 4^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (4 * n - 3) * a(n-1) - 4 * (n-1) * a(n-2) + 1.
a(n) ~ (n-1)! * exp(1/4) * 4^(n-1). - Vaclav Kotesovec, Jun 08 2022

A347012 E.g.f.: exp(x) / (1 - 4 * x)^(1/4).

Original entry on oeis.org

1, 2, 8, 64, 800, 13376, 278272, 6914048, 199629824, 6566164480, 242327576576, 9915111636992, 445432721932288, 21795710738038784, 1153805878313615360, 65700181140859518976, 4004182878034473254912, 260071258357260225609728, 17932703649301871611346944

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Binomial transform of A007696.

Cf. A000522, A007696, A056545, A084262, A088991, A346258, A347013, A347014.

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021

nmax = 18; CoefficientList[Series[Exp[x]/(1 - 4 x)^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]
Table[HypergeometricU[1/4, n + 5/4, 1/4]/Sqrt[2], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n,k) * A007696(k).

a(n) ~ n! * exp(1/4) * 4^n / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 14 2021

A277472 a(n) = (-i)^n * Integral_{x>=0} H_n(ix) exp(-x), where H_n(x) is n-th Hermite polynomial, i=sqrt(-1).

Original entry on oeis.org

1, 2, 10, 60, 492, 4920, 59160, 828240, 13253520, 238563360, 4771297440, 104968543680, 2519245713600, 65500388553600, 1834010896798080, 55020326903942400, 1760650461445075200, 59862115689132556800, 2155036164826415270400, 81891374263403780275200

Offset: 0

Vladimir Reshetnikov, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Hermite Polynomial, Incomplete Gamma Function
Wikipedia, Hermite polynomials

Cf. A056545, A277393, A277374.

FunctionExpand@Table[Exp[1/4] (-2 I)^n n! (Cos[Pi n/2] Gamma[n/2 + 1, 1/4]/Gamma[n/2 + 1] + I Gamma[(n + 1)/2, 1/4] Sin[Pi n/2]/Gamma[(n + 1)/2]), {n, 0, 20}]
FunctionExpand@Table[2^n (n!/Floor[n/2]!) Gamma[Ceiling[(n+1)/2], 1/4] Exp[1/4], {n, 0, 19}] (* Peter Luschny, Oct 19 2016 *)

for(n=0, 30, print1(round(2^n*(n!/floor(n/2)!)* incgam(ceil( (n+1)/2), 1/4)*exp(1/4)), ", ")) \\ G. C. Greubel, Jul 12 2018

def A():
    yield 1
    yield 2
    a, h, f, g, n, b = 10, 5, 1, 2, 2, False
    while True:
        yield a
        if b:
            f = h
            h = 4 * n * h + 1
            n += 1
            a = (a * h) // f
        else:
            g += 4
            a *= g
        b = not b
a = A(); print([next(a) for  in range(20)]) # _Peter Luschny, Oct 19 2016

a(n) = exp(1/4)*(-2*i)^n * n!*( cos(Pi*n/2)*Gamma(n/2 +1, 1/4)/Gamma(n/2 +1) + i*Gamma((n+1)/2, 1/4)*sin(Pi*n/2)/Gamma((n+1)/2) ).
From Peter Luschny, Oct 19 2016: (Start)
a(n) = 2^n*(n!/floor(n/2)!)*Gamma(ceiling((n+1)/2),1/4)*exp(1/4).
The swinging factorial A056040(n) divides a(n). (End)

A097821 Expansion of e.g.f. exp(2x)/(1-5x).

Original entry on oeis.org

1, 7, 74, 1118, 22376, 559432, 16783024, 587405968, 23496238976, 1057330754432, 52866537722624, 2907659574746368, 174459574484786176, 11339872341511109632, 793791063905777690624, 59534329792933326829568

Offset: 0

Paul Barry, Aug 26 2004

Second binomial transform of n!*5^n.

Robert Israel, Table of n, a(n) for n = 0..353

Cf. A056545, A000354.

f:= proc(n) option remember; 5*n*procname(n-1)+2^n end proc:
f(0):= 1:
map(f, [$0..50]); # Robert Israel, Nov 10 2022

my(x='x + O('x^25)); Vec(serlaplace(exp(2*x)/(1-5*x))) \\ Michel Marcus, Nov 08 2022

a(n) = 5*n*a(n-1) + 2^n, n > 0, a(0)=1.
D-finite with recurrence a(n) +(-5*n-2)*a(n-1) +10*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
a(n) = 5^n * n! * Sum_{k = 0..n} (2/5)^k/k! = 5^n * exp(2/5) * gamma(n + 1, 2/5). - Gerry Martens, Nov 07 2022

A375613 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], 1/4).

Original entry on oeis.org

1, 1, 5, 2, 9, 41, 6, 26, 113, 493, 24, 102, 434, 1849, 7889, 120, 504, 2118, 8906, 37473, 157781, 720, 3000, 12504, 52134, 217442, 907241, 3786745, 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861, 40320, 166320, 686160, 2831160, 11683224, 48219366, 199040786, 821723673, 3392923553

Offset: 0

Detlef Meya, Aug 21 2024

			Triangle starts:
[0] 1;
[1] 1, 5;
[2] 2, 9, 41;
[3] 6, 26, 113, 493;
[4] 24, 102, 434, 1849, 7889;
[5] 120, 504, 2118, 8906, 37473, 157781;
[6] 720, 3000, 12504, 52134, 217442, 907241, 3786745;
[7] 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861;
...

Cf. A375612, A000142, A056545 (main diagonal).

Cf. A374427, A374428, A375446, A375447, A375597, A375600.

T[n_, k_] := Sum[4^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = Sum_{j=0..k} 4^(k - j)*binomial(k, k - j)*(n - j)!.

cp's OEIS Frontend

A010845 a(n) = 3*n*a(n-1) + 1, a(0) = 1.

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A056546 a(n) = 5*n*a(n-1) + 1 with a(0)=1.

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

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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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A336807 a(n) = (n!)^2 * Sum_{k=0..n} 4^(n-k) / (k!)^2.

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A353548 Expansion of e.g.f. -log(1-4*x) * exp(x)/4.

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A347012 E.g.f.: exp(x) / (1 - 4 * x)^(1/4).

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A277472 a(n) = (-i)^n * Integral_{x>=0} H_n(i*x) * exp(-x), where H_n(x) is n-th Hermite polynomial, i=sqrt(-1).

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A010845 a(n) = 3na(n-1) + 1, a(0) = 1.

A056546 a(n) = 5na(n-1) + 1 with a(0)=1.

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

A353548 Expansion of e.g.f. -log(1-4x) exp(x)/4.

A277472 a(n) = (-i)^n * Integral_{x>=0} H_n(ix) exp(-x), where H_n(x) is n-th Hermite polynomial, i=sqrt(-1).