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A156917 General q-Narayana triangle sequence: q=3;m=2; c(n,l,m)=Product[q-binomial(n + k, l + k, m)/q-binomial(n - l + k, k, m), {k, 0, m}] 0
1, 1, 1, 1, 40, 1, 1, 1210, 1210, 1, 1, 33880, 1024870, 33880, 1, 1, 925771, 784128037, 784128037, 925771, 1, 1, 25095280, 580812061522, 16262737722616, 580812061522, 25095280, 1, 1, 678468820, 425659125229240, 325671796712891524 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 42, 2422, 1092632, 1570107618, 17424412036222, 652194913033179170,

189060566695044668933610, 188602075109681827520528645944,

1459625430842679382287597833052615968 ,...}.

I have made the general Narayana level i equal to the m =q-1 q-combination / Gaussian level,

but that is not necessary.

LINKS

Table of n, a(n) for n=0..31.

FORMULA

q=3;m=2;

c(n,l,m)=Product[q-binomial(n + k, l + k, m)/q-binomial(n - l + k, k, m), {k, 0, m}]

EXAMPLE

{1},

{1, 1},

{1, 40, 1},

{1, 1210, 1210, 1},

{1, 33880, 1024870, 33880, 1},

{1, 925771, 784128037, 784128037, 925771, 1},

{1, 25095280, 580812061522, 16262737722616, 580812061522, 25095280, 1},

{1, 678468820, 425659125229240, 325671796712891524, 325671796712891524, 425659125229240, 678468820, 1},

{1, 18326727760, 310852833944711080, 6447056947633081877440, 176165831094073962301048, 6447056947633081877440, 310852833944711080, 18326727760, 1},

{1, 494894285941, 226744821210507560554, 127139910155209947281116468, 94173897417940882107581360008, 94173897417940882107581360008, 127139910155209947281116468, 226744821210507560554, 494894285941, 1},

{1, 13362799477720, 165329327642475178468363, 2504087254915277031691820226328, 50145960006476009671341554681490292, 1359328502654886873735539664093290560, 50145960006476009671341554681490292, 2504087254915277031691820226328, 165329327642475178468363, 13362799477720, 1}

MATHEMATICA

Clear[t, n, m, i, k, c, b];

t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])]

c[n_, l_, m_] = Product[b[n + k, l + k, m]/b[n - l + k, k, m], {k, 0, m}]

Table[Flatten[Table[Table[c[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]

CROSSREFS

A001263

Sequence in context: A013375 A013419 A013420 * A176644 A078084 A037937

Adjacent sequences:  A156914 A156915 A156916 * A156918 A156919 A156920

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 18 2009

STATUS

approved

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Last modified October 20 18:08 EDT 2018. Contains 316401 sequences. (Running on oeis4.)