A255905 Expansion of exp( Sum_{n >= 1} R(n,u)*x^n/n ), where R(n,u) denotes the n-th row polynomial of A086646.

1, 1, 1, 3, 4, 1, 23, 31, 9, 1, 371, 484, 128, 16, 1, 10515, 13407, 3228, 360, 25, 1, 461869, 581680, 132291, 13260, 815, 36, 1, 28969177, 36241581, 7981991, 749199, 41167, 1603, 49, 1, 2454072147, 3058280624, 660958100, 59706312, 3060128, 106232, 2856, 64, 1

Offset: 0

Author

Peter Bala, Mar 10 2015

Keywords

• nonn
• tabl
• easy

Comments

Triangle A086646 has the e.g.f. cosh(sqrt(u)*t)/cos(t). The n-th row polynomial of A086646 is given by the formula R(n,u) = Sum_{k = 0..n} binomial(2*n,2*k)*A000364(n-k)*u^k.

It appears that in the expansion of exp( Sum_{n >= 1} R(n,u)*x^n/n ), the coefficient polynomials in u are always integer polynomials. Alternatively expressed, the o.g.f. for A086646 is (apart from its initial element) the logarithmic derivative of the o.g.f. of the present triangle.

The above conjecture can be extensively generalized. The elements of A000364 can be expressed in terms of the Euler polynomial E(n,x) as A000364(n) = (-1)^n*2^(2*n)*E(2*n,1/2). This suggests considering polynomials of the form P(n,u) = Sum_{k = 0..n} binomial(2*n,2*k)*A(n-k)*u^k, where the sequence A(n) is defined in terms of the Euler polynomials. Calculation suggests that in the expansion of exp( Sum_{n >= 1} P(n,u)*x^n/n ), the coefficient polynomials in u are always integer polynomials for the following choices of A(n):

1) A(n) := k^(2*n)*E(2*n,h/k)

2) A(n) := (4*k)^n*E(n,h/(4*k))

3) A(n) := (2*k)^(2*n+1)*E(2*n+1,h/(2*k))

In each case above, h and k are arbitrary integers except that k is nonzero.

The present triangle (up to signs) is simply the case of conjecture 1 with the choices h = 1 and k = 2.

Similar conjectures can be made if, in the above definition of the polynomial P(n,u), the factor equal to binomial(2*n,2*k) is replaced by binomial(m*n,m*k) for some fixed m = 1,2,3,....

Examples

			The triangle begins
n\k|      0      1      2     3   4   5  6
= = = = = = = = = = = = = = = = = = = = = =
0  |      1
1  |      1      1
2  |      3      4      1
3  |     23     31      9     1
4  |    371    484    128    16   1
5  |  10515  13407   3228   360  25   1
6  | 461869 581680 132291 13260 815  36  1
		

Crossrefs

Cf. A000364, A086646.

Programs

Maple

#A255905
A000364 := n -> (-1)^n*2^(2*n)*euler(2*n, 1/2):
#define row polynomials of A086646
R := proc (n, u) add(binomial(2*n, 2*k)*A000364(n-k)*u^k, k = 0 .. n) end proc:
series(exp(add(R(n, u)*x^n/n, n = 1 .. 9)), x, 9):
seq(seq(coeff(coeftayl(%, x = 0, n), u, k), k = 0 .. n), n = 0 .. 8);

Formula

O.g.f.: exp( Sum_{n >= 1} R(n,u)*x^n/n ) = exp( (1 + u)*x + (5 + 6*u + u^2)*x^2/2 + (61 + 75*u + 15*u^2 + u^3)*x^3/3 + ... ) = 1 + (1 + u)*x + (3 + 4*u + u^2)*x^2 + (23 + 31*u + 9*u^2 + u^3)*x^3 + ....