A154653 - OEIS

%I #19 Dec 26 2023 13:05:09

%S 1,1,1,1,6,1,1,15,15,1,1,45,210,210,1,1,66,495,924,66,1,1,120,1820,

%T 8008,8008,1820,1,1,153,3060,18564,43758,18564,153,1,1,231,7315,74613,

%U 646646,646646,74613,7315,1,1,378,20475,376740,13123110,30421755,30421755,13123110,376740,1

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

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

%H G. C. Greubel, <a href="/A154653/b154653.txt">Rows n = 0..100 of triangle, flattened</a>

%H A. Lakhtakia, R. Messier, V. K. Varadan, V. V. Varadan, <a href="http://dx.doi.org/10.1103/PhysRevA.34.2501">Use of combinatorial algebra for diffusion on fractals</a>, Physical Review A, volume 34, Number 3 (1986) p. 2503 (7b).

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

%e Triangle begins as:

%e   1;

%e   1,   1;

%e   1,   6,    1;

%e   1,  15,   15,     1;

%e   1,  45,  210,   210,      1;

%e   1,  66,  495,   924,     66,      1;

%e   1, 120, 1820,  8008,   8008,   1820,     1;

%e   1, 153, 3060, 18564,  43758,  18564,   153,    1;

%e   1, 231, 7315, 74613, 646646, 646646, 74613, 7315, 1;

%p 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

%t 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

%o (PARI) T(n,k) = if(k==0, 1, binomial(prime(n+1)-1, prime(k+1)-1) ); \\ _G. C. Greubel_, Dec 02 2019

%o (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

%o (Sage)

%o def T(n, k):

%o     if (k==0): return 1

%o     else: return binomial(nth_prime(n+1)-1, nth_prime(k+1)-1)

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 02 2019

%Y Cf. A154652.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Jan 13 2009

%E Edited by _G. C. Greubel_, Dec 02 2019