A329154 - OEIS

A329154

Coefficients of polynomials related to the sum of Gaussian binomial coefficients for q = 2. Triangle read by rows, T(n,k) for 0 <= k <= n.

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 26, 24, 12, 4, 1, 158, 130, 60, 20, 5, 1, 1330, 948, 390, 120, 30, 6, 1, 15414, 9310, 3318, 910, 210, 42, 7, 1, 245578, 123312, 37240, 8848, 1820, 336, 56, 8, 1, 5382862, 2210202, 554904, 111720, 19908, 3276, 504, 72, 9, 1

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OFFSET

0,4

LINKS

Table of n, a(n) for n=0..54.

FORMULA

Let P(n, k, x) = x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2) and Q(n, x) = Sum_{k=0..n} P(n-k, k, x) then T(n, k) = [x^k] Q(n, x).

T(n, k) = (1/k!) * Pochhammer(n-k+1, k) * Sum_{j=0..n-k}((-1)^j*Sum_{m=0..n-k-j} (Product_{i=1..n-k-m-j} ((2^(i+m)-1)/(2^i-1))) * binomial(n-k, j)). - Detlef Meya, Oct 07 2023

EXAMPLE

Triangle starts:

[0] [1]

[1] [1, 1]

[2] [2, 2, 1]

[3] [6, 6, 3, 1]

[4] [26, 24, 12, 4, 1]

[5] [158, 130, 60, 20, 5, 1]

[6] [1330, 948, 390, 120, 30, 6, 1]

[7] [15414, 9310, 3318, 910, 210, 42, 7, 1]

[8] [245578, 123312, 37240, 8848, 1820, 336, 56, 8, 1]

[9] [5382862, 2210202, 554904, 111720, 19908, 3276, 504, 72, 9, 1]

MAPLE

T := (n, k) -> local j, m; pochhammer(n - k + 1, k)*add((-1)^j*add(product((2^(i + m) - 1)/(2^i - 1), i = 1..n-k-m-j), m = 0..n-k-j)*binomial(n - k, j), j = 0..n-k) / k!: for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Oct 08 2023

MATHEMATICA

T[n_, k_]:= (Pochhammer[n-k+1, k]/(k!)*Sum[(-1)^j*Sum[Product[(2^(i+m)-1)/(2^i-1), {i, 1, n-k-m-j}], {m, 0, n-k-j}]*Binomial[n-k, j], {j, 0, n-k}]); Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, Oct 07 2023 *)

PROG

(Sage)

R = PolynomialRing(QQ, 'x')

x = R.gen()

@cached_function

def P(n, k, x):

if k < 0 or n < 0: return R(0)

if k == 0: return R(1)

return x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2)

def row(n): return sum(P(n-k, k, x) for k in range(n+1)).coefficients()

print(flatten([row(n) for n in range(10)]))

CROSSREFS