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A059366 Triangle T(m,s), m >= 0, 0 <= s <= m, arising in computation of certain integrals. 2
1, 1, 1, 3, 2, 3, 15, 9, 9, 15, 105, 60, 54, 60, 105, 945, 525, 450, 450, 525, 945, 10395, 5670, 4725, 4500, 4725, 5670, 10395, 135135, 72765, 59535, 55125, 55125, 59535, 72765, 135135, 2027025, 1081080, 873180, 793800, 771750, 793800, 873180 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 167, a(m,s).

Konrad Jacobs, Selecta Mathematica I (Springer 1969), Das kombinatorische arcsin-Gesetz, Lemma 3.3.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

T(m+2, s) = (2*m+3)*(T(m+1, s-1) + T(m+1, s)) - 4*(m+1)^2*T(m, s-1).

T(m, s) = Sum_{k=0..s} binomial(s, k)*binomial(2*m-2*k, s)*binomial(2*m-2*k-s, m-k).

From Reinhard Zumkeller, Apr 10 2004: (Start)

T(n,s) = A000984(s)*A000984(n-s)*A000142(n)/A000079(n).

T(n,s) = T(n,n-s).

Sum_{s=0..n} T(n,s) = A000165(n). (End)

EXAMPLE

    1;

    1,   1;

    3,   2,   3;

   15,   9,   9,  15;

  105,  60,  54,  60, 105;

  945, 525, 450, 450, 525, 945; ...

MATHEMATICA

Table[Binomial[2*s, s]*Binomial[2*n - 2*s, n - s]*n!/2^n, {n, 0, 10}, {s, 0, n}] // Flatten (* G. C. Greubel, Jan 08 2017 *)

PROG

(PARI) for(n=0, 10, for(s=0, n, print1(binomial(2*s, s)*binomial(2*n - 2*s, n - s)*n!/2^n, ", "))) \\ G. C. Greubel, Jan 08 2017

(MAGMA) /* as triangle */ [[Binomial(2*s, s)*Binomial(2*n-2*s, n-s)*Factorial(n)/2^n: s in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 09 2017

CROSSREFS

Main diagonal gives A001757.

Sequence in context: A057053 A081850 A247237 * A298594 A092950 A059239

Adjacent sequences:  A059363 A059364 A059365 * A059367 A059368 A059369

KEYWORD

tabl,nonn,easy

AUTHOR

N. J. A. Sloane, Jan 28 2001

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

STATUS

approved

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Last modified February 23 04:57 EST 2018. Contains 299473 sequences. (Running on oeis4.)