A131112 - OEIS

A131112

T(n,k) = 4*binomial(n,k) - 3*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

1, 4, 1, 4, 8, 1, 4, 12, 12, 1, 4, 16, 24, 16, 1, 4, 20, 40, 40, 20, 1, 4, 24, 60, 80, 60, 24, 1, 4, 28, 84, 140, 140, 84, 28, 1, 4, 32, 112, 224, 280, 224, 112, 32, 1, 4, 36, 144, 336, 504, 504, 336, 144, 36, 1

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OFFSET

0,2

LINKS

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

FORMULA

T(n,k) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

n-th row sum = A036563(n+2) = 2^(n+2) - 3.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 3*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021

EXAMPLE

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

4, 1;

4, 8, 1;

4, 12, 12, 1;

4, 16, 24, 16, 1;

4, 20, 40, 40, 20, 1;

...

MAPLE

seq(seq(`if`(k=n, 1, 4*binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

MATHEMATICA

Table[If[k==n, 1, 4*Binomial[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PROG

(PARI) T(n, k) = if(k==n, 1, 4*binomial(n, k)); \\ G. C. Greubel, Nov 18 2019

(Magma) [k eq n select 1 else 4*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

(Sage)

def T(n, k):

if (k==n): return 1

else: return 4*binomial(n, k)

[[T(n, k) for k in (0..n)] for n in (0..10)]

# G. C. Greubel, Nov 18 2019

(GAP)

T:= function(n, k)

if k=n then return 1;

else return 4*Binomial(n, k);

fi; end;

Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 18 2019

CROSSREFS

Cf. A007318, A036563, A131110, A131111, A131113, A131114, A131115.

Sequence in context: A072812 A244097 A162956 * A141225 A344947 A079185

Adjacent sequences: A131109 A131110 A131111 * A131113 A131114 A131115

KEYWORD

nonn,tabl,easy,less

AUTHOR

Gary W. Adamson, Jun 15 2007

STATUS

approved