A104673 -id:A104673 - cp's OEIS frontend

A062190 Coefficient triangle of certain polynomials N(5; m,x).

1, 1, 6, 1, 14, 21, 1, 24, 84, 56, 1, 36, 216, 336, 126, 1, 50, 450, 1200, 1050, 252, 1, 66, 825, 3300, 4950, 2772, 462, 1, 84, 1386, 7700, 17325, 16632, 6468, 792, 1, 104, 2184, 16016, 50050, 72072, 48048, 13728, 1287, 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002

Offset: 0

Wolfdieter Lang, Jun 19 2001

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=5) Laguerre triangle L(5; n+m,m)= A062138(n+m,m), n >= 0, is N(5; m,x)/(1-x)^(2*(m+3)), with the row polynomials N(5; m,x) := Sum_{k=0..m} a(m,k)*x^k.

			Triangle begins as:
  1;
  1,   6;
  1,  14,   21;
  1,  24,   84,    56;
  1,  36,  216,   336,    126;
  1,  50,  450,  1200,   1050,    252;
  1,  66,  825,  3300,   4950,   2772,     462;
  1,  84, 1386,  7700,  17325,  16632,    6468,    792;
  1, 104, 2184, 16016,  50050,  72072,   48048,  13728,   1287;
  1, 126, 3276, 30576, 126126, 252252,  252252, 123552,  27027,  2002;
  1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, 3003;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), this sequence (c=6).
Columns k: A028557 (k=1), A104676 (k=2), A104677 (k=3), A104678 (k=4), A104679 (k=5), A104680 (k=6).
Diagonals: A000389 (k=n), A027818 (k=n-1), A104670 (k=n-2), A104671 (k=n-3), A104672 (k=n-4), A104673 (k=n-5), A104674 (k=n-6).
Cf. A003516 (row sums), A113894 (main diagonal).

A062190:= func< n,k | Binomial(n,k)*Binomial(n+5,k) >;
[A062190(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2025

A062190 := proc(m,k)
    add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j,j=0..m) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Nov 29 2015

NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); Table[CoefficientList[NN[5, m, x], x], {m, 0, 8}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
A062190[n_,k_]:= Binomial[n,k]*Binomial[n+5,k];
Table[A062190[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 28 2025 *)

def A062190(n,k): return binomial(n,k)*binomial(n+5,k)
print(flatten([[A062190(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 28 2025

T(m, k) = [x^k]N(5; m, x), with N(5; m, x) = ((1-x)^(2*(m+3)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+6))).
N(5; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(5; m, x)= x^m*(2*m+5)! * 2F1(-m, -m; -2*m-5; (x-1)/x)/((m+5)!*m!). - Jean-François Alcover, Sep 18 2013
T(n, k) = binomial(n, k)*binomial(n+5, k). - G. C. Greubel, Feb 28 2025

A092371 Triangle read by rows: T(n, k) = binomial(n, k) * binomial(n+k, n-k).

Original entry on oeis.org

1, 6, 1, 18, 15, 1, 40, 90, 28, 1, 75, 350, 280, 45, 1, 126, 1050, 1680, 675, 66, 1, 196, 2646, 7350, 5775, 1386, 91, 1, 288, 5880, 25872, 34650, 16016, 2548, 120, 1, 405, 11880, 77616, 162162, 126126, 38220, 4320, 153, 1, 550, 22275, 205920, 630630

Offset: 1

Benoit Cloitre, Mar 20 2004

Related to the coefficients of x^k y^k in the n-th power of x^2 + x*y + 2*x + y + 1. - F. Chapoton, Jan 04 2025

			Triangle starts:
  [1]   1;
  [2]   6,    1;
  [3]  18,   15,     1;
  [4]  40,   90,    28,     1;
  [5]  75,  350,   280,    45,     1;
  [6] 126, 1050,  1680,   675,    66,    1;
  [7] 196, 2646,  7350,  5775,  1386,   91,   1;
  [8] 288, 5880, 25872, 34650, 16016, 2548, 120, 1;

First column = A002411, second column = A001297, third column = A107418, fourth column = A105251, fifth column = A104673.
Main diagonal = 1, second diagonal = A000384.
Cf. A063007, A006480 (central terms), A082759 (row sums + 1).
Cf. A104684.

T := (n, k) -> binomial(n, k) * binomial(n+k, n-k):  # Peter Luschny, Jan 04 2025

T(n,k) = binomial(n,k)*binomial(n+k,n-k)

T(n, k) = [x^(n-k)] F(-n, -n-k; 1; x). - Paul Barry, Sep 04 2008

cp's OEIS Frontend

A062190 Coefficient triangle of certain polynomials N(5; m,x).

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