A080253 -id:A080253 - cp's OEIS frontend

A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).

1, 1, 9, 73, 849, 12241, 211929, 4280473, 98806689, 2565862561, 74035143849, 2349822967273, 81361870604529, 3051889548205681, 123282485663042169, 5335770920836028473, 246332487897909570369, 12083010395805261921601, 627555570373369525058889, 34404109751876393769480073

Offset: 0

Ilya Gutkovskiy, Dec 07 2023

nonn

G. C. Greubel, Table of n, a(n) for n = 0..380

Cf. A000670, A052841, A080253, A216794, A344037, A367979, A367980, A367981, A367982, A367983.

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(2*x)) ))) // G. C. Greubel, Jun 10 2024

nmax = 19; CoefficientList[Series[Exp[-x]/(2 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

def A367977_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(2*x)) ).egf_to_ogf().list()
A367977_list(50) # G. C. Greubel, Jun 10 2024

a(n) = Sum_{k>=0} (2*k-1)^n / 2^(k+1).
a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 2^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000670(k).

A185419 Table of coefficients of a polynomial sequence of binomial type related to the enumeration of minimax trees A080795.

Original entry on oeis.org

1, 3, 1, 10, 9, 1, 42, 67, 18, 1, 248, 510, 235, 30, 1, 1992, 4378, 2835, 605, 45, 1, 19600, 44268, 34888, 10605, 1295, 63, 1, 222288, 524748, 461748, 178913, 31080, 2450, 84, 1, 2851712, 7103088, 6728428, 3069612, 690753, 77112, 4242, 108, 1

Offset: 1

Peter Bala, Feb 07 2011

DEFINITION
Define a sequence of polynomials M(n,x) by means of the recurrence relation
(1)... M(n+1,x) = x*{2*M(n,x+1)-M(n,x-1)},
with starting value M(0,x) = 1. We call these the minimax polynomials.
The first few polynomials are
M(1,x) = x
M(2,x) = x*(x   +  3)
M(3,x) = x*(x^2 +  9*x   +  10)
M(4,x) = x*(x^3 + 18*x^2 +  67*x   +  42)
M(5,x) = x*(x^4 + 30*x^3 + 235*x^2 + 510*x + 248).
This triangle lists the coefficients of these polynomials (apart from M(0,x)) in ascending powers of x.
RELATION TO MINIMAX TREES
The value M(n,1) equals the number of minimax trees on n nodes - A080795(n). This result can be used to recursively calculate the entries of A080795 - see A185420.
In addition, the minimax polynomials M(n,x) occur in the formula for the number T(n,k) of forests of k minimax trees on n nodes. ... T(n,k) = 1/k!*sum {j = 0..k} (-1)^(k-j)*binomial(k,j)*M(n,j).
ANALOGIES WITH THE MONOMIALS
{M(n,x)}n>=0 is a polynomial sequence of binomial type and so is analogous to the sequence of monomials x^n. Denoting M(n,x) by x^[n] to emphasize this analogy, we have, for example, the following analog of Bernoulli's formula for the sum of integer powers:
(2)... 1^[p]+...+(n-1)^[p] = -2*n^[p]+ 1/(p+1)*Sum_{k = 0..floor(p/2)} 8^k*binomial(p+1,2k)*B_(2k)*n^[p+1-2k], where {B_k}k>=0 = [1, -1/2, 1/6, 0, -1/30, ...] is the sequence of Bernoulli numbers.
For other polynomial sequences defined by recurrences similar to (1), and related to the zigzag numbers A000111 and the Springer numbers A001586, see A147309 and A185417, respectively. See also A185415.
The Bell transform of A143523(n). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins
n\k|.....1......2......3......4......5......6......7
====================================================
..1|.....1
..2|.....3......1
..3|....10......9......1
..4|....42.....67.....18......1
..5|...248....510....235.....30......1
..6|..1992...4378...2835....605.....45......1
..7|.19600..44268..34888..10605...1295.....63......1
..
Example of the generalized Bernoulli summation formula:
The second row of the triangle gives x^[2] = 3*x+x^2.
Then 1^[2]+2^[2]+...+(n-1)^[2] = (n^3+3*n^2-4*n)/3 = 1/3*(MB(3,n)-MB(3,0)).
From _R. J. Mathar_, Mar 15 2013: (Start)
The matrix inverse starts
       1;
      -3,       1;
      17,      -9,        1;
    -147,      95,      -18,      1;
    1697,   -1245,      305,    -30,      1;
  -24483,   19687,    -5670,    745,    -45,    1;
  423857, -365757,   118237, -18690,   1540,  -63,   1;
-8560947, 7819287, -2761122, 498197, -50190, 2842, -84, 1; (End)

Cf. A080253 (coeffs. of delta operator), A080795 (row sums), A143523 (column 1), A147309, A185415, A185417, A185420.

#A185419
M := proc(n,x) option remember;
if n = 0 then
return 1
else return
x*(2*M(n-1,x+1)-M(n-1,x-1))
end if;
end proc:
with(PolynomialTools):
for n from 1 to 10 do
CoefficientList(M(n,x),x);
end do;

M[0, ] = 1; M[n, x_] := M[n, x] = x (2 M[n-1, x+1] - M[n-1, x-1]);
Table[CoefficientList[M[n, x], x] // Rest, {n, 1, 10}] (* Jean-François Alcover, Jun 26 2019 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A143523(n), 10) # Peter Luschny, Jan 18 2016

GENERATING FUNCTION
Let a = 3-2*sqrt(2). Let f(t) = (1/2)*sqrt(2)*((1+a*exp(2*sqrt(2)*t))/ (1-a*exp(2*sqrt(2)*t))) = 1 + t + 4*t^2/2! + 20*t^3/3! + ... be the e.g.f. for A080795. Then the e.g.f. for the current table, including a constant 1, is
(1)... F(x,t) = f(t)^x = Sum_{n>=0} M(n,x)*t^n/n! = 1 + x*t + (3*x+x^2)*t^2/2! + (10*x+9*x^2+x^3)*t^3/3! + ....
ROW POLYNOMIALS
One easily checks that d/dt(F(x,t)) = x*(2*F(x+1,t)-F(x-1,t)) and hence the row generating polynomials M(n,x) satisfy the recurrence relation
(2)... M(n+1,x) = x*{2*M(n,x+1)-M(n,x-1)}.
The form of the e.g.f shows that the row polynomials are a polynomial sequence of binomial type. The associated delta operator D* is given by
(3)... D* = sqrt(2)/4*log((3+2*sqrt(2))*(sqrt(2)*exp(D)-1)/(sqrt(2)*exp(D)+1)),
where D is the derivative operator d/dx. This expands to
(4)... D* = D - 3*D^2/2! + 17*D^3/3! - 147*D^4/4! + ....
The sequence of coefficients [1,3,17,147,...] is A080253.
The delta operator D* acts as a lowering operator for the minimax polynomials
(5)...(D*) M(n,x) = n*M(n-1,x).
In what follows it will be convenient to denote M(n,x) by x^[n].
ANALOG OF THE LITTLE FERMAT THEOREM
For integer x and odd prime p
(6)... x^[p] = (-1)^((p^2-1)/8)*x (mod p).
More generally, for k = 1,2,...
(7)... x^[p+k-1] = (-1)^((p^2-1)/8)*x^[k] (mod p).
GENERALIZED BERNOULLI POLYNOMIALS ASSOCIATED WITH THE MINIMAX POLYNOMIALS
The generalized Bernoulli polynomial MB(k,x) associated with the minimax polynomial x^[k] (= M(k,x)) may be defined as the result of applying the differential operator D*/(exp(D)-1) to the polynomial x^[k]:
(8)... MB(k,x) := {D*/(exp(D)-1)} x^[k].
The first few generalized Bernoulli polynomials are
MB(0,x) = 1,
MB(1,x) = x   -  2,
MB(2,x) = x^2 -    x   + 4/3,
MB(3,x) = x^3 +  3*x^2 - 4*x,
MB(4,x) = x^4 + 10*x^3 + 3*x^2 - 14*x - 32/15.
Since exp(D)-1 is the forward difference operator it follows from (5) and (8) that
(9)... MB(k,x+1) - MB(k,x) = k*x^[k-1].
Summing (9) from x = 1 to x = n-1 and telescoping we find a closed form expression for the finite sums
(10)... 1^[p]+2^[p]+...+(n-1)^[p] = 1/(p+1)*{MB(p+1,n)-MB(p+1,1)}.
The generalized Bernoulli polynomials can be expanded in terms of the minimax polynomials x^[k]. Use (3) to express exp(D)-1 in terms of D*.
Substitute the resulting expression in (8) and expand as a power series in D* to arrive at the expansion:
(11)... MB(k,x) = -2*k*x^[k-1] + Sum_{j=0..floor(k/2)} 2^(3*j) * binomial(k,2j)*B_(2j)*x^[k-2j], where {B_j}j>=0 = [1,-1/2,1/6,0,-1/30,...] denotes the Bernoulli number sequence.
RELATION WITH OTHER SEQUENCES
Column 1 [1, 3, 10, 42, 248, ...] = A143523 with an offset of 1.
Row sums [1, 1, 4, 20, 128, 1024, ...] = A080795.

A258377 O.g.f. satisfies A^2(z) = 1/(1 - z)*( BINOMIAL(BINOMIAL(A(z))) ).

Original entry on oeis.org

1, 3, 13, 79, 649, 6955, 93813, 1539991, 29884881, 669628819, 17005862301, 482399018527, 15108642099673, 517599894435643, 19247498583665029, 771922934908235751, 33206411983713679009, 1525025984109289947171, 74466779211331635306029, 3852255519421356879419631

Offset: 0

Peter Bala, May 28 2015

For a fixed integer N, Hanna has considered the problem of finding an o.g.f. of the form A(z) = 1 + N*z + a(2)*z^2 + a(3)*z^3 + ..., with integer coefficients a(2), a(3), ... dependent on the parameter N, which is a solution to the functional equation A^(N+1) = ( BINOMIAL(A) )^N. Here BINOMIAL(F(z))= 1/(1 - z)*F(z/(1 - z)) denotes the binomial transform of the o.g.f. F(z).
The function A(z) is related to the triangle of ordered Stirling numbers A019538 via logarithmic differentiation. It can be shown that z*A'(z)/A(z) = Sum_{k >= 1} R(k,N)*z^k, where R(k,x) denotes the k-th row polynomial of A019538; equivalently, A(z) = exp( Sum_{k >= 1} R(k,N)*z^k/k ).
Cases include A084784 (N = 1), A090352 (N = 2), A090355 (N = 3), A090357 (N = 4), A090362 (N = 5) and a signed version of A084785 (N = -2).
It turns out that the o.g.f. B(z) := A(z)^(1/N) also has integer coefficients. It satisfies the functional equation B^(N+1) = BINOMIAL(B^N). For cases see A084784 (N = 1), A090351 (N = 2), A090353 (N = 3), A090356 (N = 4), A090358 (N = 5) and A084784 (N = -2).
There are similar results to the above associated with triangle A145901, which can be viewed as a type B analog of A019538.
For a fixed integer N, consider the problem of finding an o.g.f. with integer coefficients (depending on the parameter N) of the form A(z) = 1 + N*z + a(2)*z^2 + a(3)*z^3 + ..., which is a solution to the functional equation A^(N+1)(z) = 1/(1 - z) * ( BINOMIAL(BINOMIAL(A(z))) )^N; equivalently A^(N+1)(z) = 1/(1 - z)* 1/(1 - 2*z)^N*A^N(z/(1 - 2*z)). This is the type B analog of Hanna's type A functional equation above.
It can be shown that z*A'(z)/A(z) = Sum_{k >= 1} P(k,N)*z^k, where P(k,x) denotes the k-th row polynomial of A145901. However, unlike the type A situation, the type B function A(z)^(1/N) does not have all integer coefficients.
The present sequence is the case N = 1. For further examples of solutions to the type B functional equation see A258378 (N = 2), A258379 (N = 3), A258380 (N = 4) and A258381 (N = 5).
From Peter Bala, Dec 06 2017: (Start)
a(n) appears to be always odd. Calculation suggests that for k = 1,2,3,..., the sequence a(n) (mod 2^k) is purely periodic with period 2^(k-1). For example, a(n) (mod 4) = (1,3,1,3,...) seems to be purely periodic with period 2 and a(n) (mod 8) = (1,3,5,7,1,3,5,7,...) seems to be purely periodic with period 4 (both checked up to n = 1000).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..383
N. J. A. Sloane, Transforms.

Cf. A019538, A080253, A084784, A084785, A090351, A090352, A090353, A090355, A090356, A090357, A090358, A090362, A145901, A258378 (N = 2), A258379 (N = 3), A258380 (N = 4), A258381 (N = 5).

Cf. A084783.

#A258377
with(combinat):
#recursively define row polynomials R(n,x) of A145901
R := proc (n, x) option remember; if n = 0 then 1 else 1 + x*add(binomial(n, i)*2^(n-i)*R(i,x), i = 0..n-1) end if; end proc:
#define a family of sequences depending on an integer parameter k
a := proc (n, k) option remember; if n = 0 then 1 else 1/n*add(R(i+1,k)*a(n-1-i,k), i = 0..n-1) end if; end proc:
# display the case k = 1
seq(a(n,1), n = 0..19);

R[n_, x_] := R[n, x] = If[n==0, 1, 1+x*Sum[Binomial[n, i]*2^(n-i)*R[i, x], {i, 0, n-1}]];
a[n_, k_] := a[n, k] = If[n==0, 1, 1/n*Sum[R[i+1, k]*a[n-1-i, k], {i, 0, n-1}]];
a[n_] := a[n, 1];
a /@ Range[0, 19] (* Jean-François Alcover, Oct 02 2019 *)

a(0) = 1 and for n >= 1, a(n) = 1/n*Sum_{i = 0..n-1} R(i+1,1)*a(n-1-i), where R(n,x) denotes the n-th row polynomial of A145901.
O.g.f.: A(z) = 1 + 3*z + 13*z^2 + 79*z^3 + 649*z^4 + ... satisfies A^2(z) = 1/(1 - z)*1/(1 - 2*z)*A(z/(1 - 2*z)).
O.g.f.: A(z) = exp( Sum_{k >= 1} R(k,1)*z^k/k ).
1 + z*A'(z)/A(z) = 1 + 3*z + 17*z^2 + 147*z^3 + 1697*z^4 + ... is the o.g.f. for A080253.
a(n) = Sum_{j=0..n} binomial(n,j) * A084783(n,n-j). - Alois P. Heinz, Jun 09 2023
a(n) ~ (n-1)! * 2^(n - 1/2) / log(2)^(n+1). - Vaclav Kotesovec, May 28 2025

A080254 For n>3, a(n) is the number of elements in the Coxeter complex of type D_n (although the sequence starts at n=0. See comments below for precise explanation).

Original entry on oeis.org

1, 1, 9, 75, 865, 12483, 216113, 4364979, 100757313, 2616517443, 75496735057, 2396212835283, 82968104980961, 3112139513814243, 125716310807844081, 5441108944839913587, 251195548533025953409, 12321551453507301079683

Offset: 0

Paul Boddington & Tim Honeywill, Feb 10 2003

The sequence makes most sense when n>3. The values for a(2) and a(3) make sense if we regard D_2=A_1 x A_1 and D_3=A_3. The values for a(0) and a(1) have to be regarded as conventions and were included to give a nice recursive description. The corresponding sequence for type B is A080253. There one can find a worked example as well as a geometric interpretation.

Also, Eulerian D-polynomials (A066094) evaluated at 2. - Ralf Stephan, Apr 23 2004

Kenneth S. Brown, Buildings, Springer-Verlag, 1988

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Joël Gay, Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.

Cf. A000670, A080253.

CoefficientList[Series[(2*x-E^x)/(E^(2*x)-2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

a(0)=a(1)=1. For n>1, a(n)=1 + sum('2^r*binomial(n, r)*a(n-r)', 'r'=1..n)
E.g.f: (2*x-exp(x))/(exp(2*x)-2) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003
a(n) ~ n! * (sqrt(2)/log(2)-1)/2 * (2/log(2))^n. - Vaclav Kotesovec, Oct 08 2013

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 14 2003

A162312 Triangular array, inverse of 2*P - I, where P is Pascal's triangle and I is the identity matrix.

Original entry on oeis.org

1, -2, 1, 6, -4, 1, -26, 18, -6, 1, 150, -104, 36, -8, 1, -1082, 750, -260, 60, -10, 1, 9366, -6492, 2250, -520, 90, -12, 1, -94586, 65562, -22722, 5250, -910, 126, -14, 1, 1091670, -756688, 262248, -60592, 10500, -1456, 168, -16, 1, -14174522, 9825030

Offset: 0

Peter Bala, Jul 01 2009

We make a few remarks about the general array M(a) := (I - a*P)^-1, where a <> 1, and its connection with weighted sums of powers of positive integers. The present case corresponds to -M(2).
The array M(a) begins
/
| 1/(1-a)
| a/(1-a)^2............... 1/(1-a)
| (a+a^2)/(1-a)^3......... 2*a/(1-a)^2........ 1/(1-a)
| (a+4*a^2+a^3)/(1-a)^4... 3*(a+a^2)/(1-a)^3.. 3*a/(1-a)^2... 1/(1-a)
| ...
\
In the first column the numerator polynomials are the Eulerian polynomials A_n(a). See A008292.
The e.g.f. for this array is
(1)... exp(x*t)/(1-a*exp(t)) = 1/(1-a) + [a/(1-a)^2 + x/(1-a)]*t
+ [(a+a^2)/(1-a)^3 + 2*a*x/(1-a)^2 + x^2/(1-a)]*t^2/2! + ....
The row generating polynomials P_m(x) of the array M(a), which, of course, depend on a, have properties similar to those of the Bernoulli polynomials. They form an Appell sequence and may be expressed in terms of the Eulerian polynomials as
(2)... P_m(x) = sum {k=0..m} binomial(m,k) * A_k(a) / (1-a)^(k+1) * x^(m-k).
As a Newton series we have
(3)... P_m(x) = 1/(1-a)*sum {j = 0..m} sum {k = j..m}(a/(1-a))^j * k! * Stirling2(m,k) * binomial(x,k-j).
The proof of this result in the particular case a = -1 given in [Roman, p. 100] can be easily generalized to a proof of (3).
A result equivalent to (3) is
(4)... P_m(x) = 1/(1-a)*sum {j = 0..m} sum {k = 0..j} (a/(1-a))^j * (-1)^(j-k) * comb(j,k) * (x + k)^m,
which in turn leads to the infinite series expansion
(5)... P_m(x) = sum {k = 0..inf} a^k * (x + k)^m,
provided |a| < 1. See [Nelsen].
The polynomials P_m(x) satisfy the difference equation
(6)... P_m(x) - a*P_m(x + 1) = x^m (recall a <> 1),
which leads easily to the evaluation of the weighted sums of powers of integers
(7)... sum {k = 0..n-1} a^k * k^m = P_m(0) - a^n * P_m(n).
for m = 0,1,2,... and a <> 1.
More generally we have
(8)... sum {k = 0..n-1} a^k * (x + k)^m = P_m(x) - a^n * P_m(x + n).
for m = 0,1,2,... and a <> 1.
In the remaining case a = 1 the sums are evaluated in terms of the Bernoulli polynomials.
The most well-studied case is when a = -1. The row polynomials of the array M(-1) are then one half of the Euler polynomials E_m(x), which may be used to evaluate the alternating sums of powers of integers
(9)... 2*sum {k = 1..n-1} (-1)^k * k^m = E_m(0) - (-1)^n * E_m(n).

			Triangle begins
====================================================
n\k|.....0......1......2......3......4......5......6
====================================================
0..|.....1
1..|....-2......1
2..|.....6.....-4......1
3..|...-26.....18.....-6......1
4..|...150...-104.....36.....-8......1
5..|.-1082....750...-260.....60....-10......1
6..|..9366..-6492...2250...-520.....90....-12......1
...

S. Roman, The Umbral Calculus, Dover Publications.

G. F. C. De Bruyn, Formulas for a + a^2*2^p + a^3*3^p + ... + a^n*n^p, Fibonacci Quart. 33 (1995), no. 2, 98-103.
R. B. Nelsen, Problem E3062, Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377.

Cf. A000629, A000670, A007047, A008292, A080253, A154921.

#A162312
with(combinat):
T := (n,k) -> (-1)^(n+k)*binomial(n,k)
*add( j!*stirling2(n-k+1,j+1),j = 0..n):
for n from 0 to 9 do
seq(T(n,k), k = 0..n);
end do;

Table[(-1)^(n+k) Binomial[n, k] PolyLog[k-n, 1/2], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2019 *)

matrix(10, 10, n, k, 2*binomial(n-1,k-1) - (n==k))^(-1) \\ Michel Marcus, Jul 12 2018

TABLE ENTRIES
(1)... T(n,k) = (-1)^(n+k) * binomial(n,k) * A000629(n-k).
(2)... T(n,k) = (-1)^(n+k) * binomial(n,k) * sum {j = 0..n} j! * Stirling2(n-k+1,j+1).
GENERATING FUNCTION
(3)... exp(x*t)/(2*exp(t)-1) = 1 + (-2+x)*t + (6-4*x+x^2)*t^2/2!
+ ....
PROPERTIES OF ROW POLYNOMIALS
The row generating polynomials R_n(x) form an Appell sequence. The first few values are R_0(x) = 1, R_1(x) = x-2, R_2(x) = x^2-4*x+6 and R_3(x) = x^3-6*x^2+18*x-26.
They may be recursively computed by means of
(4)... R_n(x) = x^n - 2*sum {k = 0..n-1} binomial(n,k) * R_k(x).
Explicit formulas are
(5)... R_n(x) = sum {j = 0..n} sum {k = j..n} (-2)^j * k! * Stirling2(n,k) * binomial(x,k-j),
(6)... R_n(x) = (-1)^n * sum {j = 0..n} sum {k = j..n} k! * Stirling2(n,k) * binomial(-x+1,k-j),
and
(7)... R_n(x) = sum {j = 0..n} sum {k = 0..j} 2^j * (-1)^k * comb(j,k) * (x + k)^n.
Other expansions include
(8)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1)^k * A000670(k) * (x-1)^(n-k),
(9)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1/2)^k * A080253(k) * (x-1/2)^(n-k)
and
(10)... R_n(x) = sum {k = 0..n} binomial(n,k) * (-1)^k * A007047(k) * (x+1)^(n-k).
SUMS OF POWERS OF INTEGERS
The row polynomials satisfy the difference equation
(11)... 2*R_n(x+1) - R_n(x) = x^n,
and so may be used to evaluate the weighted sum of powers of integers
(12)... sum {k = 0..n-1} 2^k * k^m = 2^n*R_m(n) - R_m(0).
For example, m = 3 gives
(13)... sum {k = 0..n-1} 2^k * k^3 = 2^n*(n^3-6*n^2+18*n-26) + 26.
More generally we have
(14)... sum {k = 0..n-1} 2^k * (x + k)^m = 2^n * R_m(x + n) - R_m(x).
RELATIONS WITH OTHER SEQUENCES
(15)... Row sums [1,-1,3,-13,75,...] = (-1)^n*A000670(n).
(16)... Alt. row sums [1,-3,11,-51,299,...] = (-1)^n * A007047(n).
(17)... Column 0: (-1)^n * A000629(n).
(18)... (-2)^n * R_n(1/2) = A080253(n).
(19)... R_n(1-x) = (-1)^n * P_n(x),
where P_n(x) are the row generating polynomials of A154921.
This provides the connection between (12) and the result
(20)... sum {k = 0..n-1} (1/2)^k * k^m = 2*P_m(0) - (1/2)^(n-1) * P_m(n).

Typo corrected by Peter Bala, Nov 05 2010

A162313 Triangular array P(2I - P^2)^-1, where P is Pascal's triangle A007318 and I is the identity matrix.

Original entry on oeis.org

1, 3, 1, 17, 6, 1, 147, 51, 9, 1, 1697, 588, 102, 12, 1, 24483, 8485, 1470, 170, 15, 1, 423857, 146898, 25455, 2940, 255, 18, 1, 8560947, 2966999, 514143, 59395, 5145, 357, 21, 1, 197613377, 68487576, 11867996, 1371048, 118790, 8232, 476, 24, 1

Offset: 0

Peter Bala, Jul 01 2009

Unsigned inverse of A162315.
The row generating polynomials of this triangle may be used to evaluate the weighted sums of powers of odd numbers
(1)... 1^m + 2*3^m + 4*5^m + ... + 2^n*(2*n+1)^m
and also the sums
(2)... 1^m + (1/2)*3^m + (1/4)*5^m + ... + (1/2)^n*(2*n+1)^m.
See the Formula section below.
We make a few remarks about the general array M(a) := a*P*(I-a*P^2)^-1, where a <> 1, and its connection with weighted sums of powers of odd numbers. The present case corresponds to a = 1/2. Compare with the remarks in A162312.
The array M(a) begins
/
| a/(1-a)
| (a^2+a)/(1-a)^2 ................. a/(1-a)
| (a^3+6*a^2+a)/(1-a)^3 ........... 2*(a^2+a)/(1-a)^2 ... a/(1-a)
(a^4+23*a^3+23*a^2+a)/(1-a)^4 ...
| .
\ .
In the first column we recognize the numerator polynomials as the Eulerian polynomials of type B. See A060187.
The e.g.f. for this array is
(3)... a*exp((x+1)*t)/(1-a*exp(2*t)) = a/(1-a) +[(a^2+a)/(1-a)^2 + a/(1-a)*x]*t + [(a^3+6*a^2+a)/(1-a)^3 + 2*(a^2+a)*x/(1-a)^2 + a/(1-a)*x^2]*t^2/2! + ....
The row polynomials P_m(x), which depend on a, satisfy the difference equation
(4)... P_m(x) - a*P_m(x+2) = a*(x+1)^m.
for m >= 1.
The first few values are
P_0(x) = a/(1-a), P_1(x) = a*x/(1-a) + (a^2+a)/(1-a)^2 and
P_2(x) = a*x^2/(1-a) + 2*(a^2+a)*x/(1-a)^2 + (a^3+6*a^2+a)/(1-a)^3.
Using (4) leads to the evaluations of the weighted sums of powers of even and odd positive integers:
(5)... Sum_{k = 1..n} a^k*(2*k)^m = P_m(1) - a^n*P_m(2*n+1)
and
(6)... Sum_{k = 1..n} a^k*(2*k-1)^m = P_m(0) - a^n*P_m(2*n),
with m = 0,1,2,... and a <> 1.
In the remaining case a = 1 the sums are evaluated in terms of the Bernoulli polynomials.

			Triangle begins
  n\k|.......0.......1......2......3......4......5......6
  =======================================================
  0..|.......1
  1..|.......3.......1
  2..|......17.......6......1
  3..|.....147......51......9......1
  4..|....1697.....588....102.....12......1
  5..|...24483....8485...1470....170.....15......1
  6..|..423857..146898..25455...2940....255.....18......1
  ...

A000629, A007318, A060187, A080253 (column 0), A154921, A162312, A162314 (row sums), A162315 (unsigned inverse).

m = 8;
P = Table[Binomial[n, k], {n, 0, m}, {k, 0, m}];
T = P.Inverse[2 IdentityMatrix[m+1] - P.P];
Table[T[[n+1, k+1]], {n, 0, m}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019 *)

TABLE ENTRIES
(1)... T(n,k) = binomial(n,k)*A080253(n-k).
GENERATING FUNCTION
(2)... exp((x+1)*t)/(2-exp(2*t)) = 1 + (3+x)*t + (17+6*x+x^2)*t^2/2!
+ ....
The e.g.f. can also be written as
(3)... exp(x*t)*G(t), where G(t) = exp(t)/(2-exp(2*t)) is the e.g.f. for A080253.
ROW GENERATING POLYNOMIALS
The row generating polynomials R_n(x) form an Appell sequence. The first few values are R_0(x) = 1, R_1(x) = 3 + x, R_2(x) = 17 + 6*x + x^2 and R_3(x) = 147 + 51*x + 9*x^2 + x^3.
The row polynomials may be recursively computed by means of
(4)... R_n(x) = (x+1)^n + Sum_{k=0..n-1} 2^(n-k)*binomial(n,k)*R_k(x).
An explicit formula is
(5)... R_n(x) = Sum_{j = 0..n} Sum_{k = 0..j} (-1)^(j-k)*binomial(j,k)*(x+2*k+1)^n.
There is also a representation as an infinite series
(6)... R_n(x) = (1/2)*Sum_{k >= 0} (1/2)^k*(x+2*k+1)^n.
SUMS OF POWERS OF INTEGERS
The row polynomials satisfy the difference equation
(7)... 2*R_n(x) - R_n(x+2) = (x+1)^n,
and hence may be used to evaluate the weighted sums of powers of odd integers
(8)... Sum_{k=0..n-1} (1/2)^k*(2*k+1)^m = 2*R_m(0)-1/2^(n-1)*R_m(2*n)
as well as
(9)... Sum_{k=0..n-1} 2^k*(2*k+1)^m = (-1)^m*(2^n*R_m(-2*n)-R_m(0)).
For example, m = 2 gives
(10)... Sum_{k=0..n-1} (1/2)^k*(2*k+1)^2 = 34-2^(1-n)*(4*n^2+12*n+17)
and
(11)... Sum_{k = 0..n-1} 2^k*(2*k+1)^2 = 2^n*(4*n^2 - 12*n + 17)-17.
RELATIONS WITH OTHER SEQUENCES
(12)... Row sums = [1,4,24,208,2400,...] = 2^n*A000629(n) = A162314(n).
(13)... Column 0 = [1,3,17,147,1697,...] = A080253.
The identity
(14)... R_n(2*x-1) = 2^n*P_n(x),
where P_n(x) are the row generating polynomials of A154921, provides a surprising connection between (6) and the result
(15)... Sum_{k = 0..n-1} (1/2)^k*k^m = 2*P_m(0) - (1/2)^(n-1)*P_m(n).

A337026 a(n) = (2/3) * Sum_{k>=0} (2*k + 1)^n / 3^k.

Original entry on oeis.org

1, 2, 7, 38, 277, 2522, 27547, 351038, 5112457, 83764082, 1524907087, 30536665238, 667096092637, 15787642820042, 402374890155427, 10987722264846638, 320046586135452817, 9904844539648850402, 324568009210656076567, 11226512280285374623238

Offset: 0

Ilya Gutkovskiy, Aug 11 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A080253, A122704, A123227.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 2*Exp(x)/(3-Exp(2*x)) ))); // G. C. Greubel, Jun 09 2022

Table[2^(n + 1) HurwitzLerchPhi[1/3, -n, 1/2]/3, {n, 0, 19}]
nmax = 19; CoefficientList[Series[2 Exp[x]/(3 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!

def A337026_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2*exp(x)/(3-exp(2*x)) ).egf_to_ogf().list()
A337026_list(40) # G. C. Greubel, Jun 09 2022

E.g.f.: 2 * exp(x) / (3 - exp(2*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * A122704(k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A123227(k).
a(n) ~ n! * 2^(n+1) / (sqrt(3) * log(3)^(n+1)). - Vaclav Kotesovec, Mar 27 2022
a(n) = 1 + Sum_{k=1..n} 2^(k-1) * binomial(n,k) * a(n-k). - Seiichi Manyama, Dec 24 2023

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A162313 Triangular array P(2I - P^2)^-1, where P is Pascal's triangle A007318 and I is the identity matrix.