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

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

Row sums: A006116, first column: A135922.

Cf. A022166, A323842, A323843.

Sequence in context: A162979 A094587 A135878 * A121284 A225112 A108076

Adjacent sequences: A329151 A329152 A329153 * A329155 A329156 A329157

Keyword

nonn,tabl

Author

Vladimir Kruchinin and Peter Luschny, Mar 08 2020

Status

approved