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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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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,....

LINKS

Table of n, a(n) for n=0..44.

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 + ....

EXAMPLE

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

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);

CROSSREFS

Cf. A000364, A086646.

Sequence in context: A215241 A055133 A113084 * A055325 A162498 A134049

Adjacent sequences:  A255902 A255903 A255904 * A255906 A255907 A255908

KEYWORD

nonn,tabl,easy

AUTHOR

Peter Bala, Mar 10 2015

STATUS

approved

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Last modified January 23 22:36 EST 2020. Contains 331177 sequences. (Running on oeis4.)