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A086329 Triangle T(n,k) read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] where DELTA is the operator defined in A084938. 4
1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 11, 1, 0, 1, 16, 48, 26, 1, 0, 1, 25, 140, 202, 57, 1, 0, 1, 36, 325, 916, 747, 120, 1, 0, 1, 49, 651, 3045, 5071, 2559, 247, 1, 0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1, 0, 1, 81, 1968, 19404, 84456, 159736, 117962, 26520, 1013, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

See A087903 for another version (transposed). - Philippe Deléham, Jun 13 2004

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

Sum_{k=0..n} T(n, k) = A086211(n, 0).

T(n, 1) = 1, n > 0.

T(n, 2) = (n-1)^2, n > 0.

T(k+1, k) = 2^(k+1) - k - 2 = A000295(k+1).

Sum_{k=0..n} T(n, k) = A074664(n+1). - Philippe Deléham, Jun 13 2004

Sum_{k=0..n} T(n,k)*2^k = A171151(n). - Philippe Deléham, Dec 05 2009

T(n, k) = A087903(n, n-k+1). - G. C. Greubel, Jun 21 2022

EXAMPLE

Triangle begins:

1;

0, 1;

0, 1, 1;

0, 1, 4, 1;

0, 1, 9, 11, 1;

0, 1, 16, 48, 26, 1;

0, 1, 25, 140, 202, 57, 1;

0, 1, 36, 325, 916, 747, 120, 1;

0, 1, 49, 651, 3045, 5071, 2559, 247, 1;

0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1; ...

MATHEMATICA

T[n_, k_]:= T[n, k]= If[n==0, 1, StirlingS2[n, k] + Sum[(k-m-1)*T[n-j-1, k- m]*StirlingS2[j, m], {m, 0, k-1}, {j, 0, n-2}]];

A086329[n_, k_]:= T[n, n-k+1];

Table[A086329[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 21 2022 *)

PROG

(SageMath)

@CachedFunction

def T(n, k): # T=A087903

if (n==0): return 1

else: return stirling_number2(n, k) + sum( sum( (k-m-1)*T(n-j-1, k-m)*stirling_number2(j, m) for m in (0..k-1) ) for j in (0..n-2) )

def A086329(n, k): return T(n, n-k+1)

flatten([[A086329(n, k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Jun 21 2022

CROSSREFS

Cf. A000290, A000295, A074664, A084938, A086211, A171151.

Sequence in context: A292159 A099793 A273895 * A294118 A343648 A318996

Adjacent sequences: A086326 A086327 A086328 * A086330 A086331 A086332

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Aug 30 2003, Jun 12 2007

STATUS

approved

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Last modified December 7 06:16 EST 2022. Contains 358649 sequences. (Running on oeis4.)