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A156951
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Adjusted general Roman-Appell -Andrews q-combinations:q=3;m=2; t(n,k)=If[m == 0, n!, Product[Product[1 - (m + 1)^i, {i, 1, k}]/((1 - (m + 1))^k), {k, 1, n}]]; qR-Binomial(n,k,m)=If[n == 0, 1, t(n, m)/(t(k, m)*t(n - k, m))]
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1, 1, 1, 1, 4, 1, 1, 52, 52, 1, 1, 2080, 27040, 2080, 1, 1, 251680, 130873600, 130873600, 251680, 1, 1, 91611520, 5764196838400, 230567873536000, 5764196838400, 91611520, 1, 1, 100131391360, 2293297240551116800, 11099558644267405312000
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:
{1, 2, 6, 106, 31202, 262250562, 242096450435842, 22203703883216175640322,
1781583635462497395002916067225602,
45015382722828189618873867706650885255640985602,
8580460777569738752991223151051479368209248624034591344574054402,...}.
On page 182 of "The Umbral Calculus" Steve Roman defines:
c_n=q^-Binomial[n,2]*Product[1-q^k,{k,0,n}]/(1-q)^n
that with the inverse Binomial term out gives working q-combinations.
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REFERENCES
| Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 182
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FORMULA
| q=3;m=2; t(n,k)=If[m == 0, n!, Product[Product[1 - (m + 1)^i, {i, 1, k}]/((1 - (m + 1))^k), {k, 1, n}]];
qR-Binomial(n,k,m)=If[n == 0, 1, t(n, m)/(t(k, m)*t(n - k, m))]
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EXAMPLE
| {1},
{1, 1},
{1, 4, 1},
{1, 52, 52, 1},
{1, 2080, 27040, 2080, 1},
{1, 251680, 130873600, 130873600, 251680, 1},
{1, 91611520, 5764196838400, 230567873536000, 5764196838400, 91611520, 1},
{1, 100131391360, 2293297240551116800, 11099558644267405312000, 11099558644267405312000, 2293297240551116800, 100131391360, 1},
{1, 328430963660800, 8221562339265375772672000, 14484419666824168421262622720000, 1752614779685724378972777349120000, 14484419666824168421262622720000, 8221562339265375772672000, 328430963660800, 1},
{1, 3232089113385932800, 265379535536730646570677698560000, 511015810226219308205276356441066700800000, 22507180345603603210593191843090341770035200000, 22507180345603603210593191843090341770035200000, 511015810226219308205276356441066700800000, 265379535536730646570677698560000, 3232089113385932800, 1},
{1, 95424198983606279987200, 77104878672121712831639806271661015040000, 486992877023730754327748966963127899297274658816000000, 23443881903267084696567233422509937679937537424670851072000000, 8533573012789218829550472965793617315497263622580189790208000000, 23443881903267084696567233422509937679937537424670851072000000, 486992877023730754327748966963127899297274658816000000, 77104878672121712831639806271661015040000, 95424198983606279987200, 1}
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MATHEMATICA
| t[n_, m_] = If[m == 0, n!, Product[Product[1 - (m + 1)^i, {i, 1, k}]/((1 - (m + 1))^k), {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
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CROSSREFS
| Sequence in context: A158390 A176419 A102602 * A121066 A087565 A079163
Adjacent sequences: A156948 A156949 A156950 * A156952 A156953 A156954
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KEYWORD
| nonn,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 19 2009
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