A058942 Triangle of coefficients of Gandhi polynomials.

1, 1, 1, 2, 4, 2, 8, 22, 20, 6, 56, 184, 224, 120, 24, 608, 2248, 3272, 2352, 840, 120, 9440, 38080, 62768, 54336, 26208, 6720, 720, 198272, 856480, 1550528, 1531344, 896064, 312480, 60480, 5040, 5410688, 24719488, 48207488, 52633344, 35371776

Offset: 1

Author

David W. Wilson, Jan 12 2001

Keywords

• nonn
• tabl
• nice

Comments

(1+x)^2 divides these polynomials for n > 2. - T. D. Noe, Jan 01 2008

Examples

			Triangle starts:
[1]
[1,      1]
[2,      4,      2]
[8,      22,     20,      6]
[56,     184,    224,     120,     24]
[608,    2248,   3272,    2352,    840,    120]
[9440,   38080,  62768,   54336,   26208,  6720,   720]
[198272, 856480, 1550528, 1531344, 896064, 312480, 60480, 5040]
		

Links

• T. D. Noe, Rows n = 1..50 of triangle, flattened

Crossrefs

First column is A005439, as are row sums. See also A036970.

Cf. A084938.

Programs

Mathematica

c[1][x_] = 1; c[n_][x_] :=  c[n][x] = (x+1)*((x+1)*c[n-1][x+1] - x*c[n-1][x]); Table[ CoefficientList[ c[n][x], x], {n, 9}] // Flatten (* Jean-François Alcover, Oct 09 2012 *)

Sage

# uses[delehamdelta from A084938]
def A058942_triangle(n) :
    A = [((i+1)//2)^2 for i in (1..n)]
    B = [((i+1)//2) for i in (1..n)]
    return delehamdelta(A, B)
A058942_triangle(10) # Peter Luschny, Nov 09 2019

Formula

C_1(x) = 1; C_n(x) = (x+1)*((x+1)*C_n-1(x+1) - x*C_n-1(x)).

Triangle T(n, k), read by rows; given by [1, 1, 4, 4, 9, 9, 16, 16, 25, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 24 2005