A318180 -id:A318180 - cp's OEIS frontend

A009235 E.g.f. exp( sinh(x) / exp(x) ) = exp( (1-exp(-2*x))/2 ).

1, 1, -1, -1, 9, -23, -25, 583, -3087, 4401, 79087, -902097, 4783801, 2361049, -348382697, 4102879415, -24288551071, -47413121055, 3214104039007, -44472852461857, 326386562502889, 417716032223049, -55104307651136313, 962111031220099495

Offset: 0

R. H. Hardin

Hankel transform is (-1)^binomial(n+1,2)*A108400. - Paul Barry, Apr 15 2010

Seiichi Manyama, Table of n, a(n) for n = 0..529
I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121 [math.CO], 2001.

Cf. A004211, A317996, A318179, A318180, A318181.

a := n -> (-2)^n*add(Stirling2(n,k)*(-1/2)^k, k=0..n):
seq(a(n), n=0..23); # Peter Luschny, Jan 06 2020

With[{nn=30},CoefficientList[Series[Exp[Sinh[x]/Exp[x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Jan 07 2013 *)
Table[(-2)^n BellB[n, -1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

x='x+O('x^66); /* that many terms */
v=Vec(serlaplace(exp(sinh(x)/exp(x)))) /* Joerg Arndt, May 19 2012 */

a(n) = Sum_{k=0..n} (-2)^(n-k)*Stirling2(n, k). - Vladeta Jovovic, Apr 04 2003
From Peter Bala, May 16 2012: (Start)
Recurrence equation: a(n+1) = Sum_{k = 0..n} (-2)^(n-k)*C(n,k)*a(k). Written umbrally this is a(n+1) = (a-2)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a-2) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a+2)*(a+4)*...*(a+2*(n-1)) = 1 with a(0) = 1. Cf. A004211.
Touchard's congruence holds for odd prime p: a(p+k) = (a(k) + a(k+1)) (mod p) for k = 0,1,2, ... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) = 2 (mod p) for odd prime p. (End)
From Sergei N. Gladkovskii, Sep 21 2012 - Oct 24 2013: (Start)
Continued fractions:
G.f.: (1/E(0)-1)/x where E(k)=  1 - x/(1 - 2*x + 2*x*(k+1)/E(k+1));
G.f.: 1 +x/G(0) where G(k)= 1 + 2*x/(1 + 1/(1 + 4*x*(k+1)/G(k+1)));
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+x*2*k)/(1-x/(x-1/G(k+1)));
G.f.: 1/Q(0) where Q(k)= 1 - x/(1 + 2*x*(k+1)/Q(k+1) );
G.f.: Q(0)/(1-x), where Q(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) + (1-x+2*x*k)*(1+x+2*x*k)/Q(k+1)). (End)
Lim sup n->infinity (abs(a(n))/n!)^(1/n) / (2*abs(exp(1/LambertW(-2*n)) / LambertW(-2*n))) = 1. - Vaclav Kotesovec, Aug 04 2014
a(n) = (-2)^n*B_n(-1/2), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
G.f. A(x) satisfies: A(x) = 1 + x*A(x/(1 + 2*x))/(1 + 2*x). - Ilya Gutkovskiy, May 02 2019

Extended with signs by Olivier Gérard, Mar 15 1997

A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

Original entry on oeis.org

1, 1, -2, 1, 19, -128, 379, 1549, -32600, 261631, -845909, -10713602, 237695149, -2513395259, 11792378662, 151915180429, -4826456213273, 70741388773960, -558513179369297, -2833805536521839, 200720356696607416, -4256279445015662093, 54120395442382043743, -173423789950999240226

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..495
Eric Weisstein's World of Mathematics, Bell Polynomial

Column k=3 of A309386.

Cf. A004212, A007559, A009235, A014182, A318179, A318180, A318181.

a:=series(exp((1 - exp(-3*x))/3), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = Sum[(-3)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* Peter Luschny, Aug 20 2018 *)

a(n) = Sum_{k=0..n} (-3)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-3)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-3)^n BellPolynomial_n(-1/3). - Peter Luschny, Aug 20 2018

A318179 Expansion of e.g.f. exp((1 - exp(-4*x))/4).

Original entry on oeis.org

1, 1, -3, 5, 25, -343, 2133, -3603, -112975, 1938897, -18008275, 55198805, 1753746377, -45801271943, 649021707397, -4682002329795, -50792700319903, 2692784088681889, -59182401177647011, 801759226622986917, -2169423359710146183, -263145142263538606519, 9869607872225170545333

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..474
Eric Weisstein's World of Mathematics, Bell Polynomial

Column k=4 of A309386.

Cf. A004213, A007696, A009235, A014182, A317996, A318180, A318181.

seq(n!*coeff(series(exp((1-exp(-4*x))/4),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Exp[(1 - Exp[-4 x])/4], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-4)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = Sum[(-4)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
Table[(-4)^n BellB[n, -1/4], {n, 0, 22}] (* Peter Luschny, Aug 20 2018 *)

a(n) = Sum_{k=0..n} (-4)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-4)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-4)^n*BellPolynomial_n(-1/4). - Peter Luschny, Aug 20 2018

A318181 Expansion of e.g.f. exp((1 - exp(-6*x))/6).

Original entry on oeis.org

1, 1, -5, 19, 1, -1103, 15211, -123821, 120865, 19464193, -474727877, 7017193075, -50549088671, -931708750607, 49742453940331, -1276858353426317, 21239149342811329, -100057086073774463, -9091588769200298501, 454849803186974314579, -13529950476868715792063, 262961916344710204693681

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..447
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A005012, A008542, A009235, A014182, A317996, A318179, A318180.

seq(n!*coeff(series(exp((1-exp(-6*x))/6),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 21; CoefficientList[Series[Exp[(1 - Exp[-6 x])/6], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-6)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-6)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Table[(-6)^n BellB[n, -1/6], {n, 0, 21}] (* Peter Luschny, Aug 20 2018 *)

a(n) = Sum_{k=0..n} (-6)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-6)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-6)^n*BellPolynomial_n(-1/6). - Peter Luschny, Aug 20 2018

A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).

Original entry on oeis.org

1, 1, -1, 1, 25, -674, 15211, -331827, 5987745, 15901597, -13125035449, 1292056076070, -103145930581319, 7462324963409941, -464957409070517453, 16313974895147212801, 2059903411953959582849, -708700955022151333496910, 143215213612865558214820303, -24681846509158429152517973103

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..282
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A009235, A014182, A292866, A301419, A317996, A318179, A318180, A318181.

Table[SeriesCoefficient[Sum[x^k/Product[(1 + n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(1 - Exp[-n x])/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[(-n)^(n - k) StirlingS2[n, k], {k, n}], {n, 19}]]
Join[{1}, Table[(-n)^n BellB[n, -1/n], {n, 1, 21}]] (* Peter Luschny, Aug 20 2018 *)

{a(n) = sum(k=0, n, (-n)^(n-k)*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 27 2019

a(n) = n! * [x^n] exp((1 - exp(-n*x))/n), for n > 0.
a(n) = Sum_{k=0..n} (-n)^(n-k)*Stirling2(n,k).
a(n) = (-n)^n*BellPolynomial_n(-1/n) for n >= 1. - Peter Luschny, Aug 20 2018

A309386 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, 1, 1, 1, -3, 1, 9, 2, 1, 1, 1, -4, 5, 19, -23, -9, 1, 1, 1, -5, 11, 25, -128, -25, 9, 1, 1, 1, -6, 19, 21, -343, 379, 583, 50, 1, 1, 1, -7, 29, 1, -674, 2133, 1549, -3087, -267, 1

Offset: 0

Seiichi Manyama, Jul 27 2019

			Square array begins:
   1,  1,   1,    1,    1,    1,     1, ...
   1,  1,   1,    1,    1,    1,     1, ...
   1,  0,  -1,   -2,   -3,   -4,    -5, ...
   1, -1,  -1,    1,    5,   11,    19, ...
   1,  1,   9,   19,   25,   21,     1, ...
   1,  2, -23, -128, -343, -674, -1103, ...
   1, -9, -25,  379, 2133, 6551, 15211, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..6 give A000012, (-1)^n * A000587(n), A009235, A317996, A318179, A318180, A318181.
Rows n=0+1, 2 give A000012, A024000.
Main diagonal gives A318183.
Cf. A241578, A292861.

T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * StirlingS2[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 07 2021 *)

E.g.f. of column k: exp((1 - exp(-k*x))/k) for k > 0.

A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * A(j,k) for n > 0.

A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5x)) / (1 + 5x).

Original entry on oeis.org

1, 1, 1, -4, 16, -69, 371, -2719, 24691, -243804, 2479276, -25931249, 284075601, -3320433179, 41744590941, -561939568544, 8008026088996, -119496752915869, 1854697111334891, -29870689367146379, 499291484226079551, -8668202648905259624, 156301404533216141576

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 2 places left under 5th-order inverse binomial transform.

Cf. A010739, A051139, A318180, A350456, A351056, A351184, A351185, A351187.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 5 x)]/(1 + 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-5)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-5)^k * a(n-k-2).

cp's OEIS Frontend

A009235 E.g.f. exp( sinh(x) / exp(x) ) = exp( (1-exp(-2*x))/2 ).

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A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

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A318179 Expansion of e.g.f. exp((1 - exp(-4*x))/4).

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A318181 Expansion of e.g.f. exp((1 - exp(-6*x))/6).

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A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + n*j*x).

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A309386 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).

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A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5*x)) / (1 + 5*x).

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A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).

A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5x)) / (1 + 5x).