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A141905 A skew trinomial summed triangular sequence of coefficients: t(n,m)=Sum[n!/((n - m - k)!*m!*k!), {k, 0, m}]. 0
1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 24, 8, 1, 1, 25, 70, 40, 10, 1, 1, 36, 165, 160, 60, 12, 1, 1, 49, 336, 525, 280, 84, 14, 1, 1, 64, 616, 1456, 1120, 448, 112, 16, 1, 1, 81, 1044, 3528, 3906, 2016, 672, 144, 18, 1, 1, 100, 1665, 7680, 11970, 8064, 3360, 960, 180, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 6, 17, 50, 147, 435, 1290, 3834, 11411, 34001}.

It is obscure how the defining formula is used for the region where the sum reaches k>n-m, which needs a definition of the factorials at negative integer argument. If we trust the author's Mma implementation, Mma throws in some magic renormalization to cover these arguments. If we define, properly, t(n,m) = sum_{k=0..n-m) n!/((n-m-k)!*m!*k!), then we recover just A038207. - R. J. Mathar, Feb 07 2014

Let p(n, m, k):=n!/((n-m-k)!*m!*k!), for k<=n-m and 0<= m <=n. Let p(n, m, k):=0, for k>n-m and 0<= m <=n. It seems that t(n, m) coincides with sum_{k=0..m} p(n, m, k). - Luis Manuel Rivera Martínez, Mar 04 2014

LINKS

Table of n, a(n) for n=0..65.

FORMULA

t(n,m)=Sum[n!/((n - m - k)!*m!*k!), {k, 0, m}].

EXAMPLE

{1},

{1, 1},

{1, 4, 1},

{1, 9, 6, 1},

{1, 16, 24, 8, 1},

{1, 25, 70, 40, 10, 1},

{1, 36, 165, 160, 60, 12, 1},

{1, 49, 336, 525, 280, 84, 14, 1},

{1, 64, 616, 1456, 1120, 448, 112, 16, 1},

{1, 81, 1044, 3528, 3906, 2016, 672, 144, 18, 1},

{1, 100, 1665, 7680, 11970, 8064, 3360, 960, 180, 20, 1}

MATHEMATICA

Clear[t, n, m]; t[n_, m_] = Sum[n!/((n - m - k)!*m!*k!), {k, 0, m}]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

CROSSREFS

Sequence in context: A244811 A183153 A208513 * A114188 A110511 A082950

Adjacent sequences:  A141902 A141903 A141904 * A141906 A141907 A141908

KEYWORD

nonn,tabl,obsc,uned

AUTHOR

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

STATUS

approved

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Last modified October 1 00:54 EDT 2020. Contains 337440 sequences. (Running on oeis4.)