A154653 Triangular T(n,k) = binomial(prime(n+1) - 1, prime(k+1) - 1) with T(n,0) = 1, read by rows.

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 45, 210, 210, 1, 1, 66, 495, 924, 66, 1, 1, 120, 1820, 8008, 8008, 1820, 1, 1, 153, 3060, 18564, 43758, 18564, 153, 1, 1, 231, 7315, 74613, 646646, 646646, 74613, 7315, 1, 1, 378, 20475, 376740, 13123110, 30421755, 30421755, 13123110, 376740, 1

Offset

0,5

Comments

Row sums are: {1, 2, 8, 32, 467, 1553, 19778, 84254, 1457381, 87864065, 354929117, ...}.

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) p. 2503 (7b).

Formula

T(n,k) = binomial(prime(n+1) - 1, prime(k+1) - 1) with T(n,0) = 1.

Example

Triangle begins as:
  1;
  1,   1;
  1,   6,    1;
  1,  15,   15,     1;
  1,  45,  210,   210,      1;
  1,  66,  495,   924,     66,      1;
  1, 120, 1820,  8008,   8008,   1820,     1;
  1, 153, 3060, 18564,  43758,  18564,   153,    1;
  1, 231, 7315, 74613, 646646, 646646, 74613, 7315, 1;

Maple

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

Mathematica

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

Prog

(PARI) T(n, k) = if(k==0, 1, binomial(prime(n+1)-1, prime(k+1)-1) ); \\ G. C. Greubel, Dec 02 2019
(Magma) [k eq 0 select 1 else Binomial(NthPrime(n+1)-1, NthPrime(k+1)-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(nth_prime(n+1)-1, nth_prime(k+1)-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 02 2019

Crossrefs

Cf. A154652.

Sequence in context: A146766 A176152 A146958 * A109001 A203005 A357156

Adjacent sequences: A154650 A154651 A154652 * A154654 A154655 A154656

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 13 2009

Extensions

Edited by G. C. Greubel, Dec 02 2019

Status

approved