A094816 -id:A094816 - cp's OEIS frontend

A092553 Decimal expansion of 1/e^2.

1, 3, 5, 3, 3, 5, 2, 8, 3, 2, 3, 6, 6, 1, 2, 6, 9, 1, 8, 9, 3, 9, 9, 9, 4, 9, 4, 9, 7, 2, 4, 8, 4, 4, 0, 3, 4, 0, 7, 6, 3, 1, 5, 4, 5, 9, 0, 9, 5, 7, 5, 8, 8, 1, 4, 6, 8, 1, 5, 8, 8, 7, 2, 6, 5, 4, 0, 7, 3, 3, 7, 4, 1, 0, 1, 4, 8, 7, 6, 8, 9, 9, 3, 7, 0, 9, 8, 1, 2, 2, 4, 9, 0, 6, 5, 7, 0, 4, 8, 7, 5, 5, 0, 7, 7

Offset: 0

Mohammad K. Azarian, Apr 09 2004

Consider a substrate (such as polyvinyl alcohol or in forming the polymer of methyl vinyl ketone) in a "1,3 configuration" in which substituents branching off the substrate can irreversibly join with neighboring substituents unless the neighbor is already joined to its other neighbor. Then this constant is the fraction of joined substituents on an infinite substrate.
This also applies to reversible reactions when the rate of forward reaction is much faster than that of backward reaction; see Flory p. 1518 footnote 5. This had "satisfactory accord" with his experimental data using methyl vinyl ketone polymer for which the experimentally-obtained percentage was 0.15.
(A 1,k configuration is a substituent branching off a carbon atom, k-2 intermediate carbon atoms, and substituent branching off a carbon atom.)  - Charles R Greathouse IV, Nov 30 2012
Also the probability, as n increases without bound, that a permutation of length n is simple: no intervals of length 1 < k < n (an interval of a permutation s is a set of contiguous numbers which in s have consecutive indices). - Charles R Greathouse IV, May 14 2014

			0.1353352832366...

M. H. Albert, M. D. Atkinson and M. Klazar, The enumeration of simple permutations, J. Integer Seq. 6 (2003) 03.4.4. arXiv:math/0304213.
R. Brignall, A survey of simple permutations, Permutation Patterns, ed. S. Linton, N. Ruškuc and V. Vatter, Cambridge Univ. Press, 2010, pp. 41—65; arXiv:0801.0963.
Paul J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers, Journal of the American Chemical Society 61:6 (1939), pp. 1518-1521.
Index entries for transcendental numbers

Cf. A019774, A001113, A068985, A219863.

RealDigits[N[1/E^2,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

exp(-2) \\ Charles R Greathouse IV, Nov 30 2012

From Peter Bala, Oct 27 2019: (Start)
1/e^2 = Sum_{k >= 0} (-2)^k/k!.
This is the case n = 0 of the following series acceleration formulas:
1/e^2 = n!*2^n*Sum_{k >= 0} (-2)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*2^(n-k)*binomial(-x,k) are the (unsigned) row polynomials of A137346. Cf. A094816. (End)

A095000 E.g.f.: exp(x)/(1-x)^4.

Original entry on oeis.org

1, 5, 29, 193, 1457, 12341, 116125, 1203329, 13627073, 167525317, 2222710781, 31665408545, 482196718129, 7817359305653, 134443910166077, 2444991262876321, 46883166605035265, 945426638499719429, 20002372214708227933, 443036881445294292737, 10252840082607606694961

Offset: 0

Philippe Deléham, Jun 19 2004

nonn

Sum_{k=0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n) for x = 1, 2, 3 respectively.
From Peter Bala, Jul 10 2008: (Start)
Recurrence relation: a(0) = 1, a(1) = 5, a(n) = (n+4)*a(n-1) - (n-1)*a(n-2) for n >= 2. Let p_3(n) = n^3+2*n-1 = n^(3)-3*n^(2)+3*n^(1)-1, where n^(k) denotes the rising factorial n*(n+1)*...*(n+k-1). The polynomial p_3(n) is an example of a Poisson-Charlier polynomial c_k(x;a) at k = 3, x = -n and a = -1.
The sequence b(n) := n!*p_3(n+1) = A001565(n) satisfies the same recurrence as a(n) but with the initial conditions b(0) = 2, b(1) = 11. This leads to the finite continued fraction expansion a(n)/b(n) = 1/(2+1/(5-1/(6-2/(7-...-(n-1)/(n+4))))).
Lim_{n -> infinity} a(n)/b(n) = e/6 = 1/(2+1/(5-1/(6-2/(7-...-n/((n+5)-...))))).
a(n) = -b(n) * Sum_{k = 0..n} 1/(k!*p_3(k)*p_3(k+1)) - since the rhs satisfies the above recurrence with the same initial conditions. Hence e = -6 * Sum_{k>=0} 1/(k!*p_3(k)*p_3(k+1)).
For sequences satisfying the more general recurrence a(n) = (n+1+r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n;-1), refer to A000522 (r = 0), A001339 (r=1), A082030 (r=2) and A095177 (r=4).
{a(n)} is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial

Cf. A000522, A001339, A001341, A082030, A095177, A096307, A096341, A001565.

a := n -> hypergeom([4, -n], [], -1); seq(round(evalf(a(n), 100)), n=0..18); # Peter Luschny, Sep 20 2014

Table[n!*SeriesCoefficient[E^(x)/(1-x)^4,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

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

a(n) = Sum_{k=0..n} A094816(n, k)*4^k.
a(n) = Sum_{k=0..n} binomial(n, k)*(k+3)!/6.
a(n) ~ n!*n^3*e/6. - Vaclav Kotesovec, Oct 14 2012
a(n) = hypergeom([4, -n], [], -1). - Peter Luschny, Sep 20 2014
First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) - 1 with a(0) = 1, where P(n) = n^3 + 3*n^2 + 5*n + 2 = A001565(n). - Peter Bala, Jul 26 2021
D-finite with recurrence a(n) +(-n-4)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, Aug 01 2022
a(n) = A001341(n)/6. - Alois P. Heinz, Jan 17 2025

A095177 E.g.f.: exp(x)/(1-x)^5.

Original entry on oeis.org

1, 6, 41, 316, 2721, 25946, 271801, 3105936, 38474561, 513796366, 7360674441, 112632827396, 1833790646881, 31656637715106, 577636838177561, 11109543835539736, 224635867973671041, 4764236394052127126

Offset: 0

Philippe Deléham, Jun 20 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n), A095000(n) for x = 1, 2, 3, 4 respectively.
From Peter Bala, Jul 10 2008: (Start)
a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.
Recurrence relation: a(0) = 1, a(1) = 6, a(n) = (n+5)*a(n-1) - (n-1)*a(n-2) for n >= 2. Let p_4(n) = n^4+2*n^3+5*n^2+1 = n^(4)-4*n^(3)+6*n^(2)-4*n^(1)+1, where n^(k) denotes the rising factorial n*(n+1)*...*(n+k-1). The polynomial p_4(n) is an example of a Poisson-Charlier polynomial c_k(x;a) at k = 4, x = -n and a = -1.
The sequence b(n) := n!*p_4(n+1) = A001688(n) satisfies the same recurrence as a(n) but with the initial conditions b(0) = 9, b(1) = 53. This leads to the finite continued fraction expansion expansion a(n)/b(n) = 1/(9-1/(6-1/(7-2/(8-...-(n-1)/(n+5))))).
Lim n -> infinity a(n)/b(n) = e/24 = 1/(9-1/(6-1/(7-2/(8-...-n/((n+6)-...))))).
a(n) = b(n) * sum {k = 0..n} 1/(k!*p_4(k)*p_4(k+1)) - since the rhs satisfies the above recurrence with the same initial conditions. Hence e = 24 * sum {k = 0..inf} 1/(k!*p_4(k)p_4(k+1)).
For sequences satisfying the more general recurrence a(n) = (n+1+r)*a(n-1) - (n-1)*a(n-2), which yield series acceleration formulas for e/r! that involve the Poisson-Charlier polynomials c_r(-n;-1), refer to A000522 (r = 0), A001339 (r=1), A082030 (r=2), A095000 (r=3). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial

Cf. A000522, A001339, A082030, A095000, A096307, A096341, A094793.

CoefficientList[Series[Exp[x]/(1-x)^5, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 21 2013 *)
Table[HypergeometricPFQ[{5, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = sum(k=0,n, binomial(n, k)*(k+4)!/4! ); \\ Joerg Arndt, Apr 22 2013

a(n) = Sum_{k = 0..n} A094816(n, k)*5^k.
a(n) = Sum_{k=0..n} binomial(n, k)*(k+4)!/4!.
G.f.: 1/Q(0), where Q(k) = 1 - x - x*(k+5)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
a(n) ~ n! *exp(1)*n^4/24. - Vaclav Kotesovec, Jun 21 2013
a(n) = 2F0(5,-n;;-1). - Benedict W. J. Irwin, May 27 2016
First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) + 1 with a(0) = 1, where P(n) = n^4 + 6*n^3 + 17*n^2 + 20*n + 9 = A094793(n). - Peter Bala, Jul 26 2021

A096307 E.g.f.: exp(x)/(1-x)^6.

Original entry on oeis.org

1, 7, 55, 481, 4645, 49171, 566827, 7073725, 95064361, 1369375615, 21054430591, 344231563897, 5964569413645, 109196040092491, 2106381399472435, 42705264827626261, 907920105215691217, 20198878182718877815

Offset: 0

Philippe Deléham, Jun 26 2004

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

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

Cf. A000522, A001339, A082030, A095000, A095177, A001689, A094794.

Table[HypergeometricPFQ[{6, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)
With[{nn = 250}, CoefficientList[Series[Exp[x]/(1 - x)^6, {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094916(n, k)*6^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+5)!/5!.
a(n) = 2F0(6,-n;;-1). - Benedict W. J. Irwin, May 27 2016
From Peter Bala, Jul 25 2021: (Start)
a(n) = (n+6)*a(n-1) - (n-1)*a(n-2) with a(0) = 1 and a(1) = 7. Cf. A001689.
First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) - 1 with a(0) = 1, where P(n) = n^5 + 10*n^4 + 45*n^3 + 100*n^2 + 109*n + 44 = A094794(n).
(End)

A269953 Triangle read by rows: T(n, k) = Sum_{j=0..n} binomial(-j-1, -n-1)*S1(j, k) where S1 are the Stirling cycle numbers A132393.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -1, 2, 0, 1, 1, 0, 5, 2, 1, -1, 9, 15, 15, 5, 1, 1, 35, 94, 85, 40, 9, 1, -1, 230, 595, 609, 315, 91, 14, 1, 1, 1624, 4458, 4844, 2779, 924, 182, 20, 1, -1, 13209, 37590, 43238, 26817, 9975, 2310, 330, 27, 1

Offset: 0

Peter Luschny, Apr 12 2016

Replacing the Stirling cycle numbers in the definition by the Stirling set numbers leads to A105794.
From Wolfdieter Lang, Jun 19 2017: (Start)
The triangle t(n, k) = (-1)^(n-k)*T(n, k) is the matrix product of P = A007318 (Pascal) and s1 = A048994 (signed Stirling1). This is Sheffer (exp(t), log(1+t)).
The present triangle T is therefore the Sheffer triangle (exp(-t), -log(1-t)). Note that P is Sheffer (exp(t), t) (of the Appell type). (End)
The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=1. See A381082. - Igor Victorovich Statsenko, Feb 14 2025

			Triangle starts:
   1;
  -1,  1;
   1, -1,  1;
  -1,  2,  0,  1;
   1,  0,  5,  2,  1;
  -1,  9, 15, 15,  5,  1;
   1, 35, 94, 85, 40,  9,  1.

Peter Luschny, Extensions of the binomial
Eric Weisstein's World of Mathematics, Poisson-Charlier Polynomial

Columns k=0..4 give A033999, A002741, A381064, A381065, A381066.
Cf. A000166 (row sums), A080956 (diag n,n-1).
Cf. A105793, A105794, A132393, A007318, A048994.
Cf. A001339, A046716, A082030, A095000, A137346.
KummerU(-n,1-n-x,z): this sequence (z=-1), A094816 (z=1), |A137346| (z=2), A327997 (z=3).

A269953 := (n,k) -> add(binomial(-j-1,-n-1)*abs(Stirling1(j,k)), j=0..n):
seq(print(seq(A269953(n, k), k=0..n)), n=0..9);
# Alternative:
egf := exp(-t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # Peter Luschny, Oct 28 2019

Flatten[Table[Sum[Binomial[-j-1,-n-1] Abs[StirlingS1[j,k]], {j,0,n}], {n,0,9},{k,0,n}]]
(* Or: *)
p [n_] := HypergeometricU[-n, 1 - n - x, -1];
Table[CoefficientList[p[n], x], {n, 0, 9}] (* Peter Luschny, Oct 28 2019 *)

From Wolfdieter Lang, Jun 19 2017: (Start)
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n,k)*x^k: exp(-t)/(1 - t)^x.
E.g.f. of column k sequence: exp(-x)*(-log(1-x))^k/k!, k >= 0. (End)
From Peter Bala, Oct 26 2019: (Start)
Let R(n, x) = (-1)^n*Sum_{k >= 0} binomial(n,k)*k!* binomial(-x,k) the n-th row polynomial of this triangle.
R(n, x) = c_n(-x;-1), where c_n(x;a) denotes the n-th Poisson Charlier polynomial.
The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,.... (End)
R(n, x) = KummerU(-n, 1-n-x, -1). - Peter Luschny, Oct 28 2019

A054091 Row sums of A054090.

Original entry on oeis.org

1, 2, 4, 10, 32, 130, 652, 3914, 27400, 219202, 1972820, 19728202, 217010224, 2604122690, 33853594972, 473950329610, 7109254944152, 113748079106434, 1933717344809380, 34806912206568842, 661331331924808000, 13226626638496160002, 277759159408419360044

Offset: 0

Clark Kimberling

nonn

Row sums of A094816 as a triangular array as follows: {1}, {1,1}, {1,3}, {1,1,8}, {6,1,1,24}, {29,10,1,1,89}, ... - Michael Somos, Nov 19 2006
a(n) = (n-1)a(n-1)+2, n>0; 2=0*1+2, 4=1*2+2, 10=2*4+2, ... - Gary Detlefs, May 20 2010
Row sums of triangle A208058. - Gary W. Adamson, Feb 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings, arXiv:2501.00357 [math.CO], 2024. See p. 15.

Cf. A000522, A054090, A094816, A208058.

[n le 2 select n else (n-1)*Self(n-1) -(n-3)*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 23 2022

a:= n-> `if`( n=0, 1, add(2*(n-1)!/j!, j=0..n-1)): seq(a(n), n=0..18); # Zerinvary Lajos, Oct 20 2006
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, 2+(n-1)*a(n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 23 2022

Table[If[n==0, 1, 2*(n-1)!*Sum[1/j!, {j,0,n-1}]], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)

{a(n)= local(A); if(n<1, n==0, A=vector(n); A[1]=2; for(k=1, n-1, A[k+1]=k*A[k]+2); A[n])} /* Michael Somos, Nov 19 2006 */

{a(n)= if(n<1, n==0, n--; n!*polcoeff( 2*exp(x+x*O(x^n))/(1-x), n))} /* Michael Somos, Nov 19 2006 */

[1]+[2*factorial(n-1)*sum(1/factorial(j) for j in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jun 23 2022

a(n+1) = 2*A000522(n).
a(n+1) = Sum(2*n!/j!, j=0..n). - Zerinvary Lajos, Oct 20 2006
a(n) = 2*floor(e*(n-1)!), n>1. - Gary Detlefs, May 20 2010
a(n) = n*a(n-1) - (n-2)*a(n-2), a(0)=1, a(1)=2. - Vincenzo Librandi, Feb 23 2012

A096341 E.g.f.: exp(x)/(1-x)^7.

Original entry on oeis.org

1, 8, 71, 694, 7421, 86276, 1084483, 14665106, 212385209, 3280842496, 53862855551, 936722974958, 17205245113141, 332864226563324, 6766480571358971, 144202473398010826, 3215159679583864433

Offset: 0

Philippe Deléham, Jun 28 2004

nonn

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

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

Cf. E.g.f. exp(x)/(1-x)^k: A000522 (k = 1), A001339 (k = 2), A082030 (k = 3), A095000 (k = 4), A095177 (k = 5), A096307 (k = 6).

Cf. A094795, A094816.

Table[HypergeometricPFQ[{7, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)
With[{nn = 250}, CoefficientList[Series[Exp[x]/(1 - x)^7, {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*7^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+6)!/6!.
a(n) = 2F0(7,-n;;-1). - Benedict W. J. Irwin, May 27 2016
From Peter Bala, Jul 26 2021: (Start)
a(n) = (n+7)*a(n-1) - (n-1)*a(n-2) with a(0) = 1 and a(1) = 8.
First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) + 1 with a(0) = 1, where P(n) = n^6 + 15*n^5 + 100*n^4 + 355*n^3 + 694*n^2 + 689*n + 265 = A094795(n).
(End)

A137346 Coefficients of a special case of Poisson-Charlier polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, -2, 1, 4, -5, 1, -8, 20, -9, 1, 16, -78, 59, -14, 1, -32, 324, -360, 135, -20, 1, 64, -1520, 2254, -1165, 265, -27, 1, -128, 8336, -15232, 9954, -3045, 469, -35, 1, 256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1, -512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54

Offset: 0

Roger L. Bagula, Apr 08 2008

			{1},
{-2, 1},
{4, -5, 1},
{-8, 20, -9, 1},
{16, -78,59, -14, 1},
{-32, 324, -360, 135, -20, 1},
{64, -1520, 2254, -1165, 265, -27, 1},
{-128, 8336, -15232, 9954, -3045, 469, -35, 1},
{256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1},
{-512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54, 1},
{1024, -3492416, 8793216, -8592220,4373060, -1297569, 235473, -26370, 1770, -65, 1}

M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001) pp. 93-133.
Eric Weisstein's World of Mathematics, Poisson-Charlier Polynomial.

Cf. A046716, A092553, A094816, A159964.

R := proc(n) add((-1)^k*binomial(n,k)* k!*2^(n-k)*binomial(-x, k), k=0..n);
expand(%) end: p := n -> seq((-1)^(n-k)*coeff(R(n), x, k), k=0..n):
seq(p(n), n = 0..9);
# Or:
egf := exp(-2*t)*(1+t)^x: ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # Peter Luschny, Oct 27 2019

Ca[x, 0] = 1; Ca[x, 1] = -2 + x;
Ca[x_, n_] := Ca[x, n] = (x - n - 1) Ca[x, n - 1] - 2 (n - 1) Ca[x, n - 2];
Table[CoefficientList[Ca[x, n], x], {n, 0, 9}] // Flatten
(* The unsigned row polynomials (see Peter Bala's comment) are: *)
R[n_] := HypergeometricU[-n, 1 - n - x, 2];
Table[R[n], {n, 0, 6}] (* Peter Luschny, Oct 27 2019 *)

T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(-2*t)*(1+t)^x.
With p(0, x) = 1 and p(1, x) = x - 2 the polynomials obey the recurrence
p(n, x) = (x - n - 1)*p(n-1, x) - 2*(n - 1)*p(n-2, x).
Row sums are (-2)^n*(n-1) = (-1)^n*A159964(n-1).
From Peter Bala, Oct 23 2019: (Start)
The unsigned row polynomials are
   R(n,x) = Sum_{k=0..n} (-1)^k*binomial(n, k)*k!*2^(n-k)*binomial(-x, k).
They occur in series acceleration formulas for the constant
   1/e^2 = n!*2^n*Sum_{k >= 0}(-2)^k/(k!*R(n,k)*R(n,k+1)) = 0.1353 35283 23661 ... (cf. A092553, A046716, A094816).
(End)
R(n, x) = KummerU(-n, 1 - n - x, 2). - Peter Luschny, Oct 27 2019

Edited by Peter Luschny, Oct 27 2019

A327997 Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).

Original entry on oeis.org

1, 3, 1, 9, 7, 1, 27, 38, 12, 1, 81, 192, 101, 18, 1, 243, 969, 755, 215, 25, 1, 729, 5115, 5494, 2205, 400, 33, 1, 2187, 29322, 40971, 21469, 5355, 679, 42, 1, 6561, 187992, 323658, 209356, 66619, 11452, 1078, 52, 1, 19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1

Offset: 0

Peter Luschny, Oct 27 2019

KummerU(-n, 1-n-x, 1) are the Charlier polynomials with coefficients in A094816, the coefficients of KummerU(-n, 1-n-x, 2) are in |A137346|.

The exponential generating function of this family of sequences of polynomials is in its general form (1-t)^(-x)*exp(alpha*t) with a parameter alpha.

			The triangle starts:
      1;
      3,       1;
      9,       7,       1;
     27,      38,      12,       1;
     81,     192,     101,      18,      1;
    243,     969,     755,     215,     25,      1;
    729,    5115,    5494,    2205,    400,     33,     1;
   2187,   29322,   40971,   21469,   5355,    679,    42,    1;
   6561,  187992,  323658,  209356,  66619,  11452,  1078,   52,  1;
  19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1;

A094816 (z=1), |A137346| (z=2), this sequence (z=3).
Columns k=0..3 give A000244, A346395, A381052, A382701.
Row sums in A053486.

egf := exp(3*t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(print(n!*seq(coeff(p(n), x, k), k=0..n)), n=0..9);

p [n_] := HypergeometricU[-n, 1 - n - x, 3];
Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten

T(n, k) = sum(j=k, n, 3^(n-j)*binomial(n, j)*abs(stirling(j, k, 1))); \\ Seiichi Manyama, Apr 19 2025

T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(3*t)*(1-t)^(-x).
From Igor Victorovich Statsenko, Feb 14 2025: (Start)
T(m, n, k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), for m = -3.
The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=-3. (End)

A381022 Expansion of e.g.f. -log(1-x)^3 * exp(x) / 6.

Original entry on oeis.org

0, 0, 0, 1, 10, 75, 545, 4179, 34860, 318926, 3197210, 34975061, 415371726, 5328246417, 73470506291, 1084206640399, 17054915985752, 284945098917980, 5040033650314996, 94099409345964169, 1849525745917903666, 38176559589575462327, 825716052360614856485, 18675737859143938658251

Offset: 0

Seiichi Manyama, Feb 12 2025

nonn

Column k=3 of A094816.

Cf. A381024.

nmax=23; CoefficientList[Series[-Log[1-x]^3*Exp[x]/6, {x, 0, nmax}], x]Range[0, nmax]! (* Stefano Spezia, Feb 12 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(k, 3, 1)));

a(n) = Sum_{k=0..n} binomial(n,k) * |Stirling1(k,3)|.

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