A303987 - OEIS

A303987

Triangle read by rows: T(n, k) = (binomial(n,k)*binomial(n+k,k))^2 = A063007(n, k)^2, for n >= 0, k = 0..n.

1, 1, 4, 1, 36, 36, 1, 144, 900, 400, 1, 400, 8100, 19600, 4900, 1, 900, 44100, 313600, 396900, 63504, 1, 1764, 176400, 2822400, 9922500, 7683984, 853776, 1, 3136, 571536, 17640000, 133402500, 276623424, 144288144, 11778624, 1, 5184, 1587600, 85377600, 1200622500, 5194373184, 7070119056, 2650190400, 165636900

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

OFFSET

0,3

COMMENTS

The row sums of this triangle are b(n) = A005259(n), for n >= 0. This sequence b was used in R. Apéry's 1979 proof of the irrationality of Zeta(3). See A005259 for references and links.

Row polynomials R(n, x) := Sum_{k=0..n} T(n, k)*x^k = hypergeometric([-n, -n, n+1, n+1], [1, 1, 1], x), hence b(n) = hypergeometric([-n, -n, n+1, n+1], [1, 1, 1], 1) (see the formula in A005259 given by K. A. Penson. This is the solution to Exercise 2.14 of the Koepf reference given there, p. 29).

LINKS

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

FORMULA

T(n, k) = (binomial(n,k)*binomial(n+k,k))^2 = A063007(n, k)^2, for n >= 0 and k = 0..n.

T(n, k) = (binomial(n+k, 2*k)*cbi(k))^2, with cbi(k) = A000984(k) = binomial(2*k, k), and cbi(k)^2 = A002894(k).

G.f. for column sequences (without leading zeros):

cbi(k)^2*P2(2*k, x)/(1 - x)^(4*k+1), with the row polynomials of A008459 (Pascal entries squared) P2(2*k, x) = Sum_{j=0..2*k} A008459(2*k, j)*x^j. For a proof see the general comment in A288876 on the diagonals and columns of A008459.

EXAMPLE

The triangle T begins:

n\k 0 1 2 3 4 5 6 7 ...

0: 1

1: 1 4

2: 1 36 36

3: 1 144 900 400

4: 1 400 8100 19600 4900

5: 1 900 44100 313600 396900 63504

6: 1 1764 176400 2822400 9922500 7683984 853776

7: 1 3136 571536 17640000 133402500 276623424 144288144 11778624

----------------------------------------------------------------------------

row n = 8: 1 5184 1587600 85377600 1200622500 5194373184 7070119056 2650190400 165636900,

row n = 9: 1 8100 3920400 341510400 8116208100 63631071504 176752976400 169612185600 47869064100 2363904400,

row n = 10: 1 12100 8820900 1177862400 44188244100 572679643536 2828047622400 5446435737600 3877394192100 853369488400 34134779536.

...

MATHEMATICA

T[n_, k_] := (Gamma[k + n + 1]/(Gamma[k + 1]^2*Gamma[-k + n + 1]))^2;

Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Peter Luschny, May 14 2018 *)

PROG

(GAP) Flat(List([0..10], n->List([0..n], k->(Binomial(n, k)*Binomial(n+k, k))^2))); # Muniru A Asiru, May 15 2018

CROSSREFS

Cf. A000984, A002894, A005259, A008459, A063007.

The column sequences (without zeros) are A000012, A035287(n+1) = 4*A000217(n)^2, 36*A288876, 400*A000579(n+6)^2, 4900*A000581(n+8)^2, 63504*A001287(n+10)^2, ...

Sequence in context: A011801 A169656 A362589 * A297900 A363819 A298495

Adjacent sequences: A303984 A303985 A303986 * A303988 A303989 A303990

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, May 14 2018

STATUS

approved