A081367 -id:A081367 - cp's OEIS frontend

A046716 Coefficients of a special case of Poisson-Charlier polynomials.

1, 1, 1, 1, 3, 1, 1, 6, 8, 1, 1, 10, 29, 24, 1, 1, 15, 75, 145, 89, 1, 1, 21, 160, 545, 814, 415, 1, 1, 28, 301, 1575, 4179, 5243, 2372, 1, 1, 36, 518, 3836, 15659, 34860, 38618, 16072, 1, 1, 45, 834, 8274, 47775, 163191, 318926, 321690, 125673, 1, 1, 55, 1275, 16290, 125853, 606417, 1809905, 3197210, 2995011, 1112083, 1

Offset: 0

N. J. A. Sloane

Diagonals: A000012, A000217; A000012, A002104. - Philippe Deléham, Jun 12 2004

The sequence a(n) = Sum_{k = 0..n} T(n,k)*x^(n-k) is the binomial transform of the sequence b(n) = (n+x-1)! / (x-1)!. - Philippe Deléham, Jun 18 2004

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   8,    1;
  1, 10,  29,   24,     1;
  1, 15,  75,  145,    89,     1;
  1, 21, 160,  545,   814,   415,     1;
  1, 28, 301, 1575,  4179,  5243,  2372,     1;
  1, 36, 518, 3836, 15659, 34860, 38618, 16072,   1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

Diagonals include: A000012, A000217, A002104.

Sums include: A000522 (row), A001339, A023443 (alternating sign row), A082030, A081367.

A046716:= func< n,k | (&+[(-1)^j*Binomial(n,k-j)*StirlingFirst(j+n-k, n-k): j in [0..k]]) >;
[A046716(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2024

a := proc(n,k) option remember;
   if k = 0 then 1
elif k < 0 then 0
elif k = n then (-1)^n
else a(n-1,k) - n*a(n-1,k-1) - (n-1)*a(n-2,k-2) fi end:
A046716 := (n,k) -> abs(a(n,k));
seq(seq(A046716(n,k),k=0..n),n=0..9); # Peter Luschny, Apr 05 2011

t[, 0] = 1; t[n, k_] := (-1)^k*Sum[(-1)^i*Binomial[n, i]*StirlingS1[i, n-k], {i, n-k, n}]; Table[t[n, k] // Abs, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014 *)
T[n_, k_]:= T[n,k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, T[n-1,k] +n*T[n-1,k-1] - (n-1)*T[n-2,k-2]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 31 2024 *)

def A046716(n, k): return sum(binomial(n, k-j)*stirling_number1(j+n-k, n-k) for j in range(k+1))
flatten([[A046716(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 31 2024

Enneking and Ahuja reference gives the recurrence t(n, k) = t(n-1, k) - n*t(n-1, k-1) - (n-1)*t(n-2, k-2), with t(n, 0) = 1 and t(n, n) = (-1)^n. This sequence is T(n, k) = (-1)^k * t(n, k).
Sum_{k = 0..n} T(n, k)*x^(n-k) = A000522(n), A001339(n), A082030(n) for x = 1, 2, 3 respectively.
Sum_{k = 0..n} T(n, k)*2^k = A081367(n). - Philippe Deléham, Jun 12 2004
Let P(x, n) = Sum_{k = 0..n} T(n, k)*x^k, then Sum_{n>=0} P(x, n)*t^n / n! = exp(xt)/(1-xt)^(1/x). - Philippe Deléham, Jun 12 2004
T(n, 0) = 1, T(n, k) = (-1)^k * Sum_{i=n-k..n} (-1)^i*C(n, i)*S1(i, n-k), where S1 = Stirling numbers of first kind (A008275).
From G. C. Greubel, Jul 31 2024: (Start)
T(n, k) = T(n-1, k) + n*T(n-1, k-1) - (n-1)*T(n-2, k-2), with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^(n+1)*A023443(n). (End)

More terms from Vladeta Jovovic, Jun 15 2004

A094822 E.g.f.: exp(3x)/(1-3x)^(1/3).

Original entry on oeis.org

1, 4, 19, 118, 1021, 12088, 183727, 3389242, 73156249, 1804349548, 50009179819, 1537920654526, 51952155415381, 1911990785926432, 76137201611236999, 3261400435090171522, 149530099101901409713, 7305923490645888605908, 378947686822932957638851

Offset: 0

Philippe Deléham, Jun 12 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n) for x = 1, 2 respectively.

With[{nn=20},CoefficientList[Series[Exp[3x]/Surd[1-3x,3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 28 2018 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*3^k.
a(n) ~ Gamma(2/3)*3^(n+1/2)*n^(n-1/6)/(sqrt(2*Pi)*exp(n-1)). - Vaclav Kotesovec, Jun 15 2013
Conjecture: D-finite with recurrence: a(n) +(-3*n-1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 14 2019

Corrected and extended by Harvey P. Dale, Jul 28 2018

A094856 E.g.f.: exp(4x)/(1-4x)^(1/4).

Original entry on oeis.org

1, 5, 29, 217, 2297, 34349, 674965, 16276481, 461527793, 14993138773, 548258687501, 22272738733865, 994870668959209, 48451779617935997, 2554818339078836357, 144990720049391354449, 8811240401831517313505, 570857963393730507892901, 39275973938444154366908413

Offset: 0

Philippe Deléham, Jun 13 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n) for x = 1, 2, 3 respectively.

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

Table[n!*SeriesCoefficient[E^(4x)/(1-4x)^(1/4),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 03 2012 *)
With[{nn=20},CoefficientList[Series[Exp[4x]/(1-4x)^(1/4),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Mar 29 2013 *)

x='x+O('x^66); Vec(serlaplace(exp(4*x)/(1-4*x)^(1/4))) \\ Joerg Arndt, May 11 2013

a(n) = Sum_{k = 0..n} A046716(n, k)*4^k.
a(n) ~ n^(n-1/4)*4^n*Gamma(3/4)/(exp(n-1)*sqrt(Pi)). - Vaclav Kotesovec, Oct 03 2012
Conjectured to be D-finite with recurrence: a(n) +(-4*n-1)*a(n-1) +16*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019

Corrected and extended by Harvey P. Dale, Mar 29 2013

A094869 E.g.f.: exp(5x)/(1-5x)^(1/5).

Original entry on oeis.org

1, 6, 41, 356, 4401, 78826, 1893481, 56341416, 1978638881, 79749105326, 3622010623401, 182895318578956, 10160561511881041, 615728464210461906, 40414538467581457001, 2855999961062529064976, 216180544920721807887681

Offset: 0

Philippe Deléham, Jun 16 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n) for x = 1, 2, 3, 4 respectively.

With[{nn=20},CoefficientList[Series[Exp[5x]/(1-5x)^(1/5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2014 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*5^k.
Conjectured to be D-finite with recurrence: a(n) +(-5*n-1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * 5^n * n^(n - 3/10) / (Gamma(1/5) * exp(n-1)). - Vaclav Kotesovec, Nov 19 2021

A094905 Expansion of e.g.f.: exp(6x)/(1-6x)^(1/6).

Original entry on oeis.org

1, 7, 55, 541, 7585, 157231, 4452247, 157484725, 6594785281, 317357589655, 17222102537911, 1039632137764237, 69073193451776545, 5007661199176196671, 393324947394545293975, 33268708968518818629541

Offset: 0

Philippe Deléham, Jun 16 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n) for x = 1, 2, 3, 4, 5 respectively.

Cf. A046716.

Cf. A000522, A081367, A094822, A094856, A094869.

With[{nn=20},CoefficientList[Series[Exp[6x]/Surd[1-6x,6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 15 2022 *)

E.g.f.: exp(6*x)/(1-6*x)^(1/6).
a(n) = Sum_{k = 0..n} A046716(n, k)*6^k.
Conjectured to be D-finite with recurrence: a(n) +(-6*n-1)*a(n-1) +36*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * 3^n * n^(n - 1/3) / (Gamma(1/6) * exp(n - 1)). - Vaclav Kotesovec, Nov 19 2021

A295099 a(n) = n! * [x^n] exp(nx)/sqrt(1 - 2x).

Original entry on oeis.org

1, 2, 11, 96, 1145, 17320, 317547, 6843872, 169603793, 4752704160, 148631984075, 5132717953792, 194022218612169, 7969667589513344, 353510496652374635, 16842274069331520000, 857827370723082312737, 46516913938434654949888, 2675772791433589181094027, 162742831545094476694814720

Offset: 0

Ilya Gutkovskiy, Nov 14 2017

nonn

The n-th term of the n-th binomial transform of A001147.

G. C. Greubel, Table of n, a(n) for n = 0..250
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A001147, A063170, A081367, A084262, A295098, A295100.

Table[n! SeriesCoefficient[Exp[n x]/Sqrt[1 - 2 x], {x, 0, n}], {n, 0, 19}]

a(n) ~ 2^(n+1) * n^n / exp(n/2). - Vaclav Kotesovec, Nov 14 2017

A094911 Expansion of e.g.f. exp(7x)/(1-7x)^(1/7).

Original entry on oeis.org

1, 8, 71, 778, 12125, 284012, 9241891, 378595022, 18409947641, 1029827400400, 64998958518719, 4565303338264082, 353016345110857429, 29793105387299603252, 2724646021507044539675, 268374407984059193374678

Offset: 0

Philippe Deléham, Jun 17 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n) for x = 1, 2, 3, 4, 5, 6 respectively.

Cf. A046716, A000522, A081367, A094822, A094856, A094869, A094905.

my(x='x+O('x^20)); Vec(serlaplace(exp(7*x)/(1-7*x)^(1/7))) \\ Michel Marcus, Jan 23 2023

a(n) = Sum_{k = 0..n} A046716(n, k)*7^k.
Conjectured to be D-finite with recurrence: a(n) +(-7*n-1)*a(n-1) +49*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 7^n / (Gamma(1/7) * exp(n-1) * n^(6/7)). - Vaclav Kotesovec, Nov 19 2021

Corrected by D. S. McNeil, Aug 20 2010

A094935 E.g.f.: exp(8x)/(1-8x)^(1/8).

Original entry on oeis.org

1, 9, 89, 1073, 18321, 476473, 17484457, 813648417, 45054110369, 2872362067433, 206710159889529, 16558892507010961, 1460688620617834801, 140655075719488236057, 14678730623948132120009

Offset: 0

Philippe Deléham, Jun 18 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n) for x = 1, 2, 3, 4, 5, 6, 7 respectively.

With[{nn=20},CoefficientList[Series[Exp[8x]/Surd[1-8x,8],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 25 2019 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*8^k.
D-finite with recurrence: a(n) +(-8*n-1)*a(n-1) +64*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 8^n / (Gamma(1/8) * exp(n-1) * n^(7/8)). - Vaclav Kotesovec, Nov 19 2021

A095176 E.g.f.: exp(9x)/(1-9x)^(1/9).

Original entry on oeis.org

1, 10, 109, 1432, 26497, 754894, 30787885, 1603546156, 99602138593, 7128277455538, 576063289419661, 51832424202980320, 5136461847251936929, 555721381650431686582, 65167921144448534609677

Offset: 0

Philippe Deléham, Jun 20 2004

nonn

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n), A094935(n) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.
From Vaclav Kotesovec, Nov 19 2021: (Start)
In general, for k > 0, if e.g.f. = exp(k*x) / (1 - k*x)^(1/k), then a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * k^n / (Gamma(1/k) * exp(n-1) * n^(1 - 1/k)).
Equivalently, a(n) ~ n! * exp(1) * k^n / (Gamma(1/k) * n^(1 - 1/k)). (End)

With[{nn=20},CoefficientList[Series[Exp[9x]/Surd[1-9x,9],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 25 2020 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*9^k.
D-finite with recurrence a(n) +(-9*n-1)*a(n-1) +81*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 9^n / (Gamma(1/9) * exp(n-1) * n^(8/9)). - Vaclav Kotesovec, Nov 19 2021

cp's OEIS Frontend

A046716 Coefficients of a special case of Poisson-Charlier polynomials.

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A094822 E.g.f.: exp(3x)/(1-3x)^(1/3).

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A094856 E.g.f.: exp(4x)/(1-4x)^(1/4).

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A094869 E.g.f.: exp(5x)/(1-5x)^(1/5).

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A094905 Expansion of e.g.f.: exp(6*x)/(1-6*x)^(1/6).

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A295099 a(n) = n! * [x^n] exp(n*x)/sqrt(1 - 2*x).

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A094911 Expansion of e.g.f. exp(7*x)/(1-7*x)^(1/7).

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A094935 E.g.f.: exp(8x)/(1-8x)^(1/8).

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A094905 Expansion of e.g.f.: exp(6x)/(1-6x)^(1/6).

A295099 a(n) = n! * [x^n] exp(nx)/sqrt(1 - 2x).

A094911 Expansion of e.g.f. exp(7x)/(1-7x)^(1/7).