0,6

Row sums are:

{1, 2, 4, 11, 93, 20180, 194220318, 119325331254865, 9324327722816579893293,

186569521829486603544960896563978,

2217289090251397995170754246987585930272602436,...}.

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.

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 182

q=4;m=3; t(n,k)=If[m == 0, n!, Product[Product[1 - (m + 1)^i, {i, 1, k}]/((1 - (m + 1))^k), {k, 1, n}]];

{1},

{1, 1},

{1, 1, 2},

{1, 1, 3, 6},

{1, 1, 4, 63, 24},

{1, 1, 5, 208, 19845, 120},

{1, 1, 6, 525, 432640, 193786425, 720},

{1, 1, 7, 1116, 4685625, 108886835200, 119216439727875, 5040},

{1, 1, 8, 2107, 32381856, 14260348265625, 9975288480661504000, 9314352420075537699375, 40320},

{1, 1, 9, 3648, 164259613, 733821300216576, 59241410073722830078125, 998839514786016848710205440000, 185570682255459175888706396203125, 362880},

{1, 1, 10, 5913, 665395200, 19912576089143185, 64954781923091107431960576, 1343980039782222974663472614501953125, 328049824383657403926718014214742474752000000, 1889239264523760551396858476166367156828515625, 3628800}

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

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

a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];

b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}];

Flatten[%]

Sequence in context: A123353 A156540 A156582 this_sequence A156881 A056646 A056056

Adjacent sequences: A156950 A156951 A156952 this_sequence A156954 A156955 A156956

nonn,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 19 2009