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A181144 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows. 6
1, 1, 1, 1, 9, 1, 1, 36, 36, 1, 1, 100, 419, 100, 1, 1, 225, 2699, 2699, 225, 1, 1, 441, 12138, 35052, 12138, 441, 1, 1, 784, 42865, 286206, 286206, 42865, 784, 1, 1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1, 1, 2025, 330903, 7958563 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Compare g.f. to that of the following triangle variants:

* Pascal's: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)*y^k] * x^n/n );

* Narayana: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n );

* A181143: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n );

* A218115: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5*y^k] * x^n/n );

* A218116: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6*y^k] * x^n/n ).

LINKS

Table of n, a(n) for n=0..48.

EXAMPLE

G.f.: A(x,y) = 1 + (1+y)*x + (1+9*y+y^2)*x^2 + (1+36*y+36*y^2+y^3)*x^3 + (1+100*y+419*y^2+100*y^3+y^4)*x^4 +...

The logarithm of the g.f. equals the series:

log(A(x,y)) = (1 + y)*x

+ (1 + 2^4*y + y^2)*x^2/2

+ (1 + 3^4*y + 3^4*y^2 + y^3)*x^3/3

+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4

+ (1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5)*x^5/5 +...

Triangle begins:

1;

1, 1;

1, 9, 1;

1, 36, 36, 1;

1, 100, 419, 100, 1;

1, 225, 2699, 2699, 225, 1;

1, 441, 12138, 35052, 12138, 441, 1;

1, 784, 42865, 286206, 286206, 42865, 784, 1;

1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1;

1, 2025, 330903, 7958563, 36955542, 36955542, 7958563, 330903, 2025, 1;

1, 3025, 776688, 31205941, 261852055, 525079969, 261852055, 31205941, 776688, 3025, 1; ...

Note that column 1 forms the sum of cubes (A000537), and forms the squares of the triangular numbers.

Inverse binomial transform of columns begins:

[1];

[1, 8, 19, 18, 6];

[1, 35, 348, 1549, 3713, 5154, 4161, 1818, 333];

[1, 99, 2500, 27254, 161793, 589819, 1409579, 2282850, 2529900, 1893972, 917349, 259854, 32726]; ...

PROG

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^4*y^j)*x^m/m)+O(x^(n+1))), n, x), k, y)}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A000537 (column 1), A166992 (row sums), A166898 (antidiagonal sums), A218140.

Cf. variants: A001263 (Narayana), A181143, A218115, A218116.

Sequence in context: A171822 A176490 A174158 * A142468 A304321 A156278

Adjacent sequences:  A181141 A181142 A181143 * A181145 A181146 A181147

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 13 2010

STATUS

approved

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Last modified October 19 22:28 EDT 2018. Contains 316378 sequences. (Running on oeis4.)