A129063 -id:A129063 - cp's OEIS frontend

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 26, 36, 12, 1, 0, 150, 250, 120, 20, 1, 0, 1082, 2040, 1230, 300, 30, 1, 0, 9366, 19334, 13650, 4270, 630, 42, 1, 0, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 0, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1

Offset: 0

Wolfdieter Lang, May 04 2007

Matrix product of Stirling2 with unsigned Stirling1 triangle.
For the subtriangle without column no. m=0 and row no. n=0 see A079641.
The reversed matrix product |S1|. S2 is given in A111596.
As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
E.g.f. for row polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n) is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
A048993*A132393 as infinite lower triangular matrices. - Philippe Deléham, Nov 01 2009
Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011.
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle begins:
  1;
  0,    1;
  0,    2,    1;
  0,    6,    6,    1;
  0,   26,   36,   12,   1;
  0,  150,  250,  120,  20,  1;
  0, 1082, 2040, 1230, 300, 30,  1;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First ten rows and more.

Columns k=0..3 give A000007, A000629(n-1), A129063, A129064.

Cf. A000629, A000670, A005649, A079641, A129062, A325872, A325873, A383140.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n,1/2), 9); # Peter Luschny, Jan 27 2016

rows = 9;
t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* Peter Luschny, Jun 27 2019 *)

def a_row(n):
    s = sum(stirling_number2(n,k)*rising_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

a(n,m) = Sum_{k=m..n} S2(n,k) * |S1(k,m)|, n>=0; S2=A048993, S1=A048994.
E.g.f. of column k (with leading zeros): (f(x)^k)/k! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
Sum_{0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - Philippe Deléham, Nov 19 2011

New name by Peter Luschny, Jun 27 2019

A180875 Sum_{j>=1} j^n2^j/binomial(2j,j) = r_n*Pi/2 + s_n with integer r_n and s_n; sequence gives s_n.

Original entry on oeis.org

1, 3, 11, 55, 355, 2807, 26259, 283623, 3473315, 47552791, 719718067, 11932268231, 215053088835, 4186305575415, 87534887434835, 1956680617267879, 46561960552921315, 1175204650272267479, 31357650670190565363, 881958890078887314567, 26078499305918584929155, 808742391638178302137783

Offset: 0

Jonathan Vos Post, Sep 23 2010

In the references, the infinite series is S_n(2) = A014307(n+1)*Pi/2 + A180875(n) for n >= 1 (and S_0(2) is not defined). - Petros Hadjicostas, May 14 2020

Seiichi Manyama, Table of n, a(n) for n = 0..423
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011; see Table IV on p. 14.
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130; see Table 2.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
Feng Qi and Mark Daniel Ward, Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function, arXiv:2110.08576 [math.CO], 2021.

The values of r_n give A014307.

Cf. A000629, A129063.

f := n -> sum(j^n*(j!)^2*2^j/(2*j)!, j = 1..infinity):
seq(f(n), n = 0..5); # gives
# [1+(1/2)*Pi, 3+Pi, 11+(7/2)*Pi, 55+(35/2)*Pi, 355+113*Pi, 2807+(1787/2)*Pi].

Table[Expand[FunctionExpand[FullSimplify[Sum[j^n*2^j/Binomial[2*j, j], {j, 1, Infinity}]]]][[1]], {n, 0, 20}] (* Vaclav Kotesovec, May 14 2020 *)

N=20; x='x+O('x^N); f=sqrt(exp(x)/(2-exp(x))); Vec(serlaplace(deriv(f*intformal(f)))) \\ Seiichi Manyama, Oct 22 2019

# An alternative version of the sequence starts (for n >= 0):
# 0, 1, 3, 11, ..., or in terms of the approximation: [(1/2)*Pi, 1+(1/2)*Pi,
# 3+Pi, 11+(7/2)*Pi, ...]. Similar to the formula of Detlef Meya above, the
# sequence then can be computed (without a special initial case) as:
from functools import cache
from math import comb as binomial
@cache
def a(n): return n + sum((binomial(n, j) - 1) * a(n - j) for j in range(1, n))
print([a(n) for n in range(23)])  # Peter Luschny, Jun 09 2023

a(0)=1; if n>=1, then a(n) = a(n-1) + 1 + Sum_{m=1..n} binomial(n,m)*a(n-m). - Detlef Meya, Jan 22 2018
E.g.f.: 2*(arcsin(exp(x/2)/sqrt(2)) - Pi/4) * sqrt(exp(x)/(2-exp(x))^3) + exp(x)/(2-exp(x)). - Seiichi Manyama, Oct 21 2019
a(n) ~ Pi * n^(n+1) / (sqrt(2) * exp(n) * (log(2))^(n + 3/2)). - Vaclav Kotesovec, Oct 22 2019
E.g.f.: d/dx (f(x) * Integral f(x) dx), where f(x) = sqrt(exp(x)/(2-exp(x))), cf. A014307. - Seiichi Manyama, Oct 22 2019

Attribution corrected by M. Lawrence Glasser, Sep 25 2010
Provided a better definition following a suggestion from Herb Conn. - N. J. A. Sloane, Feb 08 2011
Missing a(15) inserted by Seiichi Manyama, Oct 20 2019

cp's OEIS Frontend

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A180875 Sum_{j>=1} j^n2^j/binomial(2j,j) = r_n*Pi/2 + s_n with integer r_n and s_n; sequence gives s_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A129064 Fourth column (m=3) of triangle A129062 and third column of triangle A079641.

Views

Author

Keywords

Crossrefs

Formula

cp's OEIS Frontend

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A180875 Sum_{j>=1} j^n*2^j/binomial(2*j,j) = r_n*Pi/2 + s_n with integer r_n and s_n; sequence gives s_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A129064 Fourth column (m=3) of triangle A129062 and third column of triangle A079641.

Views

Author

Keywords

Crossrefs

Formula

A180875 Sum_{j>=1} j^n2^j/binomial(2j,j) = r_n*Pi/2 + s_n with integer r_n and s_n; sequence gives s_n.