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A218115 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5 * 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. 4
1, 1, 1, 1, 17, 1, 1, 98, 98, 1, 1, 354, 2251, 354, 1, 1, 979, 23803, 23803, 979, 1, 1, 2275, 158367, 617036, 158367, 2275, 1, 1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1, 1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1, 1, 15333 (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 );

* A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*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..46.

EXAMPLE

G.f.: A(x,y) = 1 + (1+y)*x + (1+17*y+y^2)*x^2 + (1+98*y+98*y^2+y^3)*x^3 + (1+354*y+2251*y^2+354*y^3+y^4)*x^4 +...

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

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

+ (1 + 2^5*y + y^2)*x^2/2

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

+ (1 + 4^5*y + 6^5*y^2 + 4^5*y^3 + y^4)*x^4/4

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

Triangle begins:

1;

1, 1;

1, 17, 1;

1, 98, 98, 1;

1, 354, 2251, 354, 1;

1, 979, 23803, 23803, 979, 1;

1, 2275, 158367, 617036, 158367, 2275, 1;

1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1;

1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1;

1, 15333, 10057620, 575963523, 4066874561, 4066874561, 575963523, 10057620, 15333, 1; ...

Note that column 1 forms the sum of fourth powers (A000538).

PROG

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^5*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. A000538 (column 1), A218117 (row sums).

Cf. variants: A001263 (Narayana), A181143, A181144, A218116.

Sequence in context: A103637 A229956 A157274 * A144442 A157151 A176794

Adjacent sequences:  A218112 A218113 A218114 * A218116 A218117 A218118

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 21 2012

STATUS

approved

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Last modified September 27 12:01 EDT 2020. Contains 337380 sequences. (Running on oeis4.)