0,5

Row sums are:

{1, 2, 7, 212, 205277, 10871114102, 219992119346624627,

1092725218416585412236327752, 30515253650844374162858001929528333214377,

863340390368883195792691921471708513645438894959386505002,

215000329605050439646427912760212760884667834881314160940622513473238063082812

7,...}.

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}]];

qR-Binomial(n,k,m)=If[n == 0, 1, t(n, m)/(t(k, m)*t(n - k, m))]

{1},

{1, 1},

{1, 5, 1},

{1, 105, 105, 1},

{1, 8925, 187425, 8925, 1},

{1, 3043425, 5432513625, 5432513625, 3043425, 1},

{1, 4154275125, 2528644954460625, 214934821129153125, 2528644954460625, 4154275125, 1},

{1, 22686496457625, 18849189581462430815625, 546343760018688557190890625, 546343760018688557190890625, 18849189581462430815625, 22686496457625, 1},

{1, 495586515116818125, 2248624343928882581174778890625, 88965754070982873104954068787605078125, 30337322138205159728789337456573331640625, 88965754070982873104954068787605078125, 2248624343928882581174778890625, 495586515116818125, 1},

{1, 43304845277422684580625, 4292259471742181452395960623410004765625, 927393612389393841019723221446191045365104103515625, 431669267790829204210245469270451314930868181542900390625, 431669267790829204210245469270451314930868181542900390625, 927393612389393841019723221446191045365104103515625, 4292259471742181452395960623410004765625, 43304845277422684580625, 1},

{1, 15136126045591163828042953125, 131093519300778600932325391900230129176225431640625, 618744575092116216638049133513159824648766221714500244509033203125, 1572789536246720488159542717561568743217333450840645955311078780953369140625, 214685771697677346633777580947154133449166016039748172899962253600134887695312 5, 157278953624672048815954271756156874321733345084064595531107878095336914062 5, 618744575092116216638049133513159824648766221714500244509033203125, 131093519300778600932325391900230129176225431640625, 15136126045591163828042953125, 1}

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}]

Sequence in context: A156691 A111820 A106238 this_sequence A158748 A086039 A097413

Adjacent sequences: A156949 A156950 A156951 this_sequence A156953 A156954 A156955

nonn,tabl,uned

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