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A131113 T(n,k) = 5*binomial(n,k) - 4*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n). 6
1, 5, 1, 5, 10, 1, 5, 15, 15, 1, 5, 20, 30, 20, 1, 5, 25, 50, 50, 25, 1, 5, 30, 75, 100, 75, 30, 1, 5, 35, 105, 175, 175, 105, 35, 1, 5, 40, 140, 280, 350, 280, 140, 40, 1, 5, 45, 180, 420, 630, 630, 420, 180, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).

LINKS

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

FORMULA

T(n,k) = 5*A007318(n,k) - 4*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 4*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:

  1;

  5,  1;

  5, 10,  1;

  5, 15, 15,  1;

  5, 20, 30,  20,  1;

  5, 25, 50,  50, 25,  1;

  5, 30, 75, 100, 75, 30, 1;

  ...

MAPLE

seq(seq(`if`(k=n, 1, 5*binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

MATHEMATICA

Table[If[k==n, 1, 5*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, 5*binomial(n, k)); \\ G. C. Greubel, Nov 18 2019

(MAGMA) [k eq n select 1 else 5*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 5*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 5*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, A048487, A131110, A131112, A131114, A131115.

Sequence in context: A170903 A319663 A255166 * A139426 A143384 A046611

Adjacent sequences:  A131110 A131111 A131112 * A131114 A131115 A131116

KEYWORD

nonn,tabl,easy,less

AUTHOR

Gary W. Adamson, Jun 15 2007

STATUS

approved

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Last modified January 18 09:26 EST 2022. Contains 350454 sequences. (Running on oeis4.)