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A118976
Triangle read by rows: T(n,k) = binomial(n-1,k-1)*binomial(n,k-1)/k + binomial(n-1,k)*binomial(n,k)/(k+1) (1 <= k <= n). In other words, to each entry of the Narayana triangle (A001263) add the entry on its right.
2
1, 2, 1, 4, 4, 1, 7, 12, 7, 1, 11, 30, 30, 11, 1, 16, 65, 100, 65, 16, 1, 22, 126, 280, 280, 126, 22, 1, 29, 224, 686, 980, 686, 224, 29, 1, 37, 372, 1512, 2940, 2940, 1512, 372, 37, 1, 46, 585, 3060, 7812, 10584, 7812, 3060, 585, 46, 1, 56, 880, 5775, 18810, 33264, 33264, 18810, 5775, 880, 56, 1
OFFSET
1,2
COMMENTS
Sum of entries in row n = 2*Cat(n)-1, where Cat(n) are the Catalan numbers (A000108).
Row sums = A131428 starting (1, 3, 9, 27, 83, ...). - Gary W. Adamson, Aug 31 2007
FORMULA
G.f.: A001263(x, y)*(x + x*y) + x*y. - Vladimir Kruchinin, Oct 21 2020
EXAMPLE
First few rows of the triangle:
1;
2, 1;
4, 4, 1;
7, 12, 7, 1;
11, 30, 30, 11, 1;
16, 65, 100, 65, 16, 1;
...
Row 4 of the triangle = (7, 12, 7, 1), derived from row 4 of the Narayana triangle, (1, 6, 6, 1): = ((1+6), (6+6), (6+1), (1)).
MAPLE
T:=(n, k)->binomial(n-1, k-1)*binomial(n, k-1)/k+binomial(n-1, k) *binomial(n, k)/ (k+1): for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
# Alternatively:
gf := 1 - ((1/2)*(x + 1)*(sqrt((x*y + y - 1)^2 - 4*y^2*x) + x*y + y - 1))/(y*x):
sery := series(gf, y, 10): coeffy := n -> expand(coeff(sery, y, n)):
seq(print(seq(coeff(coeffy(n), x, k), k=1..n)), n=1..8); # Peter Luschny, Oct 21 2020
MATHEMATICA
With[{B=Binomial}, Table[B[n-1, k-1]*B[n, k-1]/k + B[n-1, k]*B[n, k]/(k+1), {n, 12}, {k, n}]//Flatten] (* G. C. Greubel, Aug 12 2019 *)
PROG
(PARI) T(n, k) = b=binomial; b(n-1, k-1)*b(n, k-1)/k + b(n-1, k)*b(n, k)/(k+1);
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 12 2019
(Magma) B:=Binomial; [B(n-1, k-1)*B(n, k-1)/k + B(n-1, k)*B(n, k)/(k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 12 2019
(Sage)
def T(n, k):
b=binomial
return b(n-1, k-1)*b(n, k-1)/k + b(n-1, k)*b(n, k)/(k+1)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Aug 12 2019
(GAP) B:=Binomial; Flat(List([1..12], n-> List([1..n], k-> B(n-1, k-1)*B(n, k-1)/k + B(n-1, k)*B(n, k)/(k+1) ))); # G. C. Greubel, Aug 12 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 07 2006
EXTENSIONS
Edited by N. J. A. Sloane, Nov 29 2006
STATUS
approved