A277660 -id:A277660 - cp's OEIS frontend

A387341 Expansion of 1/(1 - 6*x + x^2)^(5/2).

1, 15, 155, 1365, 10990, 83538, 610050, 4325310, 29979015, 204086025, 1369220853, 9075850875, 59550467340, 387359772660, 2500864350900, 16040872988748, 102298452571965, 649077104453715, 4099652984281855, 25788295829930865, 161619907171129946, 1009512779437342950

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A002802, A387313, A387343.

Cf. A277660.

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 6*x + x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/(1-6*x+x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/(1-6*x+x^2)^(5/2))

n*a(n) = 3*(2*n+3)*a(n-1) - (n+3)*a(n-2) for n > 1.
a(n) = (binomial(n+4,2)/6) * A387337(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/6)^(n-2*k) * binomial(-5/2,k) * binomial(k,n-k).
a(n) = A277660(n+2)/2.
a(n) ~ n^(3/2) * (1 + sqrt(2))^(2*n+5) / (3*2^(17/4)*sqrt(Pi)). - Vaclav Kotesovec, Aug 27 2025

A376467 Triangular array read by rows: A063007 * A007318.

Original entry on oeis.org

1, 3, 2, 13, 18, 6, 63, 132, 90, 20, 321, 900, 930, 420, 70, 1683, 5910, 8190, 5600, 1890, 252, 8989, 37926, 65940, 60480, 30870, 8316, 924, 48639, 239624, 501228, 577080, 395010, 160776, 36036, 3432, 265729, 1497096, 3660300, 5072760, 4358970, 2378376, 804804, 154440, 12870, 1462563, 9274410, 25951860, 42060480, 43513470, 29801772, 13513500, 3912480, 656370, 48620

Offset: 0

Peter Bala, Sep 30 2024

Note that the n-th row generating polynomial of A063007 is equal to P(n,2*x + 1), where P(n,x) denotes the n-th Legendre polynomial.

The matrix product A063007 * A007318^(-1) is equal to a signed version of A063007 and A007318^(-1) * A063007 = A115951.

			Triangle begins
 n\k|     0       1       2       3       4       5      6     7
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |     1
  1 |     3       2
  2 |    13      18       6
  3 |    63     132      90      20
  4 |   321     900     930     420      70
  5 |  1683    5910    8190    5600    1890     252
  6 |  8989   37926   65940   60480   30870    8316    924
  7 | 48639  239624  501228  577080  395010  160776  36036  3432
  ...

A000984 (main diagonal), A002457( (1/3)*first subdiagonal ), A001850 (Column 0), A002695 ( (1/2)*Column 1 ), A277660 ( (1/3)*Column 2 ), A006442 (row sums).

Cf. A063007, A007318, A115951, A118384.

A376467 := proc(n, k); add(binomial(n, j)*binomial(n+j, j)*binomial(j, k), j = k..n) end:
seq(print(seq(A376467(n, k) , k = 0..n)), n = 0..10);

T(n, k) = Sum_{j = k..n} binomial(n, j)*binomial(n+j, j)*binomial(j, k).
(n - k)*T(n, k) = 3*(2*n - 1)*T(n-1, k) - (n + k - 1)*T(n-2, k).
T(n, k) = (1/k!) * (d/dx)^k (P(n, 2*x+1)) evaluated at x = 1, where P(n,x) denotes the n-th Legendre polynomial.
G.f. for triangle: 1/sqrt(1 - 6*t + t^2 - 4*t*x) = 1 + (3 + 2*x)*t + (13 + 18*x + 6*x^2)*t^2 + ....
G.f. for column k: binomial(2*k, k) * x^k/(1 - 6*x + x^2)^(k+1/2).
T(n, k) is divisible by binomial(2*k, k) and the array ( T(n, k)/binomial(2*k, k) )n,k >= 0 is the Riordan array (1/sqrt(1 - 6*x + x^2), x/(1 - 6*x + x^2)).
T(n, k) is divisible by binomial(n+k, k) and the array ( T(n, k)/binomial(n+k, k) )n,k >= 0 is the Riordan array A118384.
T(n, n) = binomial(2*n, n); T(n, n-1) = 3*(2*n-1)!/(n-1)!^2 = 3 * A002457(n-1) for n >= 1.
The n-th row polynomial R(n, x) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k, k)*(1 + x)^k = P(n, 2*x+3) = hypergeom([-n, n+1], [1], -1-x).
Recurrence: n*R(n, x) = (2*x + 3)*(2*n - 1)*R(n-1, x) - (n - 1)*R(n-2, x) with R(0, x) = 1.
If we set R(-1,x) = 1, we can run the recurrence backwards to give R(-n, x) = Sum_{k = 0..n} binomial(-n, k)*binomial(-n+k, k)*(1 + x)^k = R(n-1, x).
R(n, x) = (-1)^n * R(n, -x-3).
R(n, x) = 1/n! * (d/dx)^n( ((1 + x)*(2 + x))^n ).
R(1, x) = 3 + 2*x divides R(2*n+1, x) in the polynomial ring Z[x].

cp's OEIS Frontend

A387341 Expansion of 1/(1 - 6*x + x^2)^(5/2).

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