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A156715 Symmetrical version of Graham and Riordan 1966:k=1; 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
3, 3, 3, 12, 3, 3, 25, 25, 3, 3, 42, 90, 42, 3, 3, 63, 231, 231, 63, 3, 3, 88, 490, 840, 490, 88, 3, 3, 117, 918, 2394, 2394, 918, 117, 3, 3, 150, 1575, 5796, 8820, 5796, 1575, 150, 3, 3, 187, 2530, 12474, 26796, 26796, 12474, 2530, 187, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums are:

{6, 18, 56, 180, 594, 2002, 6864, 23868, 83980,...}.

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..53.

FORMULA

k=1;

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

{3, 3},

{3, 12, 3},

{3, 25, 25, 3},

{3, 42, 90, 42, 3},

{3, 63, 231, 231, 63, 3},

{3, 88, 490, 840, 490, 88, 3},

{3, 117, 918, 2394, 2394, 918, 117, 3},

{3, 150, 1575, 5796, 8820, 5796, 1575, 150, 3},

{3, 187, 2530, 12474, 26796, 26796, 12474, 2530, 187, 3}

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: A274993 A233202 A101480 * A133797 A225073 A152575

Adjacent sequences:  A156712 A156713 A156714 * A156716 A156717 A156718

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 14 2009

STATUS

approved

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Last modified June 14 15:44 EDT 2021. Contains 345025 sequences. (Running on oeis4.)