A141617 - OEIS

A141617

Triangle read by rows: T(n, k) = binomial(n,k)*prime(k)*prime(n-k), for 1 <= k <= n-1, n >= 1, with T(0, 0) = 1, T(n, 0) = T(n, n) = prime(n).

1, 2, 2, 3, 8, 3, 5, 18, 18, 5, 7, 40, 54, 40, 7, 11, 70, 150, 150, 70, 11, 13, 132, 315, 500, 315, 132, 13, 17, 182, 693, 1225, 1225, 693, 182, 17, 19, 272, 1092, 3080, 3430, 3080, 1092, 272, 19, 23, 342, 1836, 5460, 9702, 9702, 5460, 1836, 342, 23

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

OFFSET

0,2

COMMENTS

For the purpose of this sequence define prime(0)=1.

LINKS

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

FORMULA

Symmetry: T(n, k) = T(n, n-k).

EXAMPLE

Triangle begins as:

2, 2;

3, 8, 3;

5, 18, 18, 5;

7, 40, 54, 40, 7;

11, 70, 150, 150, 70, 11;

13, 132, 315, 500, 315, 132, 13;

17, 182, 693, 1225, 1225, 693, 182, 17;

19, 272, 1092, 3080, 3430, 3080, 1092, 272, 19;

23, 342, 1836, 5460, 9702, 9702, 5460, 1836, 342, 23;

29, 460, 2565, 10200, 19110, 30492, 19110, 10200, 2565, 460, 29;

...

MAPLE

p:= n-> `if`(n=0, 1, ithprime(n)):

T:= (n, k)-> binomial(n, k)*p(k)*p(n-k):

seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Apr 26 2023

MATHEMATICA

A141617[n_, k_]:= If[n==0, 1, If[k==0 || k==n, Prime[n], Binomial[n, k]*Prime[k]*Prime[n-k]]];

Table[A414617[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

PROG

(Magma)

function A141617(n, k)

if n eq 0 then return 1;

else return Binomial(n, k)*NthPrime(k)*NthPrime(n-k);

end if;

end function;

[A141617(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 26 2024

(SageMath)

def A141617(n, k):

if n==0: return 1

elif k==0 or k==n: return nth_prime(n)

else: return binomial(n, k)*nth_prime(k)*nth_prime(n-k)

flatten([[A141617(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 26 2024

CROSSREFS

Cf. A008578, A098350.

Sequence in context: A135835 A177696 A134574 * A267644 A204197 A238654

Adjacent sequences: A141614 A141615 A141616 * A141618 A141619 A141620

KEYWORD

nonn,easy,tabl,changed

AUTHOR

Roger L. Bagula and Gary W. Adamson, Aug 23 2008

STATUS

approved