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A156716 Symmetrical version of Graham and Riordan 1966:k=2; t(n,m,k)=(((2*k + 1)/(m + k + 1))*Binomial[n - 1 - k, m - k]*Binomial[n + k, m + k] + ((2*k + 1)/( n - m - 1 + k + 1))*Binomial[n - 1 - k, n - m - 1 - k]*Binomial[ n + k, n - m - 1 + k]) 0
5, 0, 5, 5, 15, 15, 5, 5, 35, 70, 35, 5, 5, 60, 210, 210, 60, 5, 5, 90, 486, 840, 486, 90, 5, 5, 125, 960, 2550, 2550, 960, 125, 5, 5, 165, 1705, 6435, 9900, 6435, 1705, 165, 5, 5, 210, 2805, 14245, 31185, 31185, 14245, 2805, 210, 5 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums are:

{10, 40, 150, 550, 2002, 7280, 26520, 96900,...}.

First triangle sequence in k is twice A001263 ( Narayana numbers).

Successive k terms shift over one so that the early terms are zero.

A symmetrical/ reverse coefficient/ toral transform was used to make the result a symmetrical sequence.

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p.17

Graham,R. L. and J. Riordan,The solution of a certain recurrence, Amer. Masth. Monthly 73, 1966,pp. 604-608

LINKS

Table of n, a(n) for n=0..51.

FORMULA

k=2;

t(n,m,k)=(((2*k + 1)/(m + k + 1))*Binomial[n - 1 - k, m - k]*Binomial[n + k, m + k] +

((2*k + 1)/( n - m - 1 + k + 1))*Binomial[n - 1 - k, n - m - 1 - k]*Binomial[ n + k, n - m - 1 + k])

EXAMPLE

{5, 0, 5},

{5, 15, 15, 5},

{5, 35, 70, 35, 5},

{5, 60, 210, 210, 60, 5},

{5, 90, 486, 840, 486, 90, 5},

{5, 125, 960, 2550, 2550, 960, 125, 5},

{5, 165, 1705, 6435, 9900, 6435, 1705, 165, 5},

{5, 210, 2805, 14245, 31185, 31185, 14245, 2805, 210, 5}

MATHEMATICA

Clear[t, n, m, k];

t[n_, m_, k_] = (((2*k + 1)/(m + k + 1))*Binomial[ n - 1 - k, m - k]*Binomial[n + k, m + k] + ((2*k + 1)/(n - m - 1 + k + 1))*Binomial[n - 1 - k, n - m - 1 - k]*Binomial[n + k, n - m - 1 + k]);

Table[Table[Table[t[n, m, k], {m, 0, n - 1}], {n, k + 1, 10}], {k, 0, 9}];

Table[Flatten[Table[Table[t[n, m, k], {m, 0, n - 1}], {n, k + 1, 10}]], {k, 0, 9}]

CROSSREFS

A001263

Sequence in context: A091672 A144702 A238192 * A055510 A200397 A265302

Adjacent sequences:  A156713 A156714 A156715 * A156717 A156718 A156719

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 14 2009

STATUS

approved

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Last modified June 20 00:16 EDT 2021. Contains 345154 sequences. (Running on oeis4.)