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A156950 Adjusted general Roman-Appell -Andrews q-combinations:q=2;m=1; 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))] 0
1, 1, 1, 1, 3, 1, 1, 21, 21, 1, 1, 315, 2205, 315, 1, 1, 9765, 1025325, 1025325, 9765, 1, 1, 615195, 2002459725, 30036895875, 2002459725, 615195, 1, 1, 78129765, 16021680259725, 7450081320772125, 7450081320772125, 16021680259725 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 5, 44, 2837, 2070182, 34043045717, 14932206158323232,

501603204672964021000277, 983025583835307758960367512885402,

540587942385612482647357191397895843519061077,...}.

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..33.

FORMULA

q=2;m=1; 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))]

EXAMPLE

{1},

{1, 1},

{1, 3, 1},

{1, 21, 21, 1},

{1, 315, 2205, 315, 1},

{1, 9765, 1025325, 1025325, 9765, 1},

{1, 615195, 2002459725, 30036895875, 2002459725, 615195, 1},

{1, 78129765, 16021680259725, 7450081320772125, 7450081320772125, 16021680259725, 78129765, 1},

{1, 19923090075, 518862115211194125, 15200065665111931891875, 471202035618469888648125, 15200065665111931891875, 518862115211194125, 19923090075, 1},

{1, 10180699028325, 67610327922594650458125, 251541858674536106216688028125, 491261249991369015441191718928125, 491261249991369015441191718928125, 251541858674536106216688028125, 67610327922594650458125, 10180699028325, 1},

{1, 10414855105976475, 35343501752520121310936218125, 33531036618160892148938540775439171875, 8316736543439064240593375207072953239328125, 523954402236661047157382638045596054077671875, 8316736543439064240593375207072953239328125, 33531036618160892148938540775439171875, 35343501752520121310936218125, \10414855105976475, 1}

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}]

CROSSREFS

Sequence in context: A111382 A173884 A176418 * A083998 A277170 A216922

Adjacent sequences:  A156947 A156948 A156949 * A156951 A156952 A156953

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 19 2009

STATUS

approved

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Last modified November 21 11:35 EST 2019. Contains 329370 sequences. (Running on oeis4.)