A154652 Triangle read by rows, T(n, k) = binomial(3*(prime(n+1) - 1)/2, 3*(prime(k+1) - 1)/2) with T(n,0) = 1.

1, 1, 1, 1, 20, 1, 1, 84, 84, 1, 1, 455, 5005, 5005, 1, 1, 816, 18564, 48620, 816, 1, 1, 2024, 134596, 1307504, 1307504, 134596, 1, 1, 2925, 296010, 4686825, 17383860, 4686825, 2925, 1, 1, 5456, 1107568, 38567100, 1037158320, 1037158320, 38567100, 1107568, 1

Offset

0,5

Comments

Row sums are {1, 2, 22, 170, 10467, 68818, 2886226, 27059372, 2153671434, 905636138220, 7579946523936, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

A. Lakhtakia, R. Messier, V. K. Varadan, V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) page 2503 (7b).

Formula

T(n, k) = binomial(3*(prime(n+1) - 1)/2, 3*(prime(k+1) - 1)/2) with T(n,0) = 1.

Example

Triangle begins as:
  1;
  1,    1;
  1,   20,      1;
  1,   84,     84,       1;
  1,  455,   5005,    5005,        1;
  1,  816,  18564,   48620,      816,       1;
  1, 2024, 134596, 1307504,  1307504,  134596,    1;
  1, 2925, 296010, 4686825, 17383860, 4686825, 2925, 1;

Maple

seq(seq( `if`(k=0, 1, binomial(3*(ithprime(n+1)-1)/2, 3*(ithprime(k+1)-1)/2) ), k=0..n), n=0..10); # G. C. Greubel, Dec 02 2019

Mathematica

T[n_, k_]:= If[k==0, 1, Binomial[3*(Prime[n+1] -1)/2, 3*(Prime[k+1] -1)/2]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(PARI) T(n, k) = if(k==0, 1, binomial(3*(prime(n+1)-1)/2, 3*(prime(k+1)-1)/2) ); \\ G. C. Greubel, Dec 02 2019
(Magma) [k eq 0 select 1 else Round( Gamma((3*NthPrime(n+1)-1)/2)/( Gamma((3*NthPrime(k+1)-1)/2)*Gamma((3*NthPrime(n+1)-3*NthPrime(k+1))/2 + 1) ) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 02 2019
(Sage)
def T(n, k):
    if (k==0): return 1
    else: return binomial(3*(nth_prime(n+1)-1)/2, 3*(nth_prime(k+1)-1)/2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 02 2019

Crossrefs

Cf. A154653.

Sequence in context: A040401 A040400 A139459 * A155516 A174674 A144443

Adjacent sequences: A154649 A154650 A154651 * A154653 A154654 A154655

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 13 2009

Extensions

Edited by G. C. Greubel, Dec 02 2019

Status

approved