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A173567 Triangle T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = (1/2)*Sum_{j=1..2*n} k^j, read by rows. 1
2, 5, 5, 9, 30, 9, 14, 123, 123, 14, 20, 425, 1092, 425, 20, 27, 1413, 7650, 7650, 1413, 27, 35, 4872, 54051, 87380, 54051, 4872, 35, 44, 17783, 426573, 943190, 943190, 426573, 17783, 44, 54, 67875, 3655854, 12192579, 12207030, 12192579, 3655854, 67875, 54 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

FORMULA

T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = (1/2)*Sum_{j=1..2*n} k^j.

T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = k*(1 - k^(2*n))/(1-k) with f(n, 1) = 2*n. - G. C. Greubel, Apr 25 2021

EXAMPLE

Triangle begins as:

2;

5, 5;

9, 30, 9;

14, 123, 123, 14;

20, 425, 1092, 425, 20;

27, 1413, 7650, 7650, 1413, 27;

35, 4872, 54051, 87380, 54051, 4872, 35;

44, 17783, 426573, 943190, 943190, 426573, 17783, 44;

54, 67875, 3655854, 12192579, 12207030, 12192579, 3655854, 67875, 54;

MATHEMATICA

f[n_, k_]:= If[k==1, 2*n, k*(1-k^(2*n))/(1-k)];

T[n_, k_]:= (f[k, n-k+1] + f[n-k+1, k])/2;

Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 25 2021 *)

PROG

(Sage)

def f(n, k): return 2*n if k==1 else k*(1-k^(2*n))/(1-k)

def T(n, k): return (f(k, n-k+1) + f(n-k+1, k))/2

flatten([[T(n, k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Apr 25 2021

CROSSREFS

Sequence in context: A050175 A243333 A059797 * A288726 A344572 A265129

Adjacent sequences: A173564 A173565 A173566 * A173568 A173569 A173570

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 22 2010

EXTENSIONS

Edited by G. C. Greubel, Apr 25 2021

STATUS

approved

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Last modified December 2 11:09 EST 2022. Contains 358493 sequences. (Running on oeis4.)