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Search: id:A125814
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| A125814 |
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q-Bell numbers for q=4; eigensequence of A022168, which is the triangle of Gaussian binomial coefficients [n,k] for q=4. |
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+0
6
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| 1, 1, 2, 8, 72, 1552, 84416, 12107584, 4726583424, 5150624868864, 16010990175691264, 144648776120641766400, 3857411545088966609514496, 307705704204270334224705015808, 74294186209325019487040708053442560
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) for n>0, with a(0)=1.
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EXAMPLE
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The recurrence: a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k)
is illustrated by:
a(2) = 1*(1) + 5*(1) + 1*(2) = 8;
a(3) = 1*(1) + 21*(1) + 21*(2) + 1*(8) = 72;
a(4) = 1*(1) + 85*(1) + 357*(2) + 85*(8) + 1*(72) = 1552.
Triangle A022168 begins:
1;
1, 1;
1, 5, 1;
1, 21, 21, 1;
1, 85, 357, 85, 1;
1, 341, 5797, 5797, 341, 1;
1, 1365, 93093, 376805, 93093, 1365, 1; ...
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PROGRAM
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(PARI) /* q-Binomial coefficients: */ {C_q(n, k)=if(n<k|k<0, 0, if(n==0|k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j)))} /* q-Bell numbers = eigensequence of q-binomial triangle: */ {B_q(n)=if(n==0, 1, sum(k=0, n-1, B_q(k)*C_q(n-1, k)))} /* Eigensequence at q=4: */ {a(n)=subst(B_q(n), q, 4)}
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CROSSREFS
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Cf. A022168; A125810, A125811, A125812, A125813, A125815.
Sequence in context: A038057 A005615 A107270 this_sequence A013002 A012998 A064605
Adjacent sequences: A125811 A125812 A125813 this_sequence A125815 A125816 A125817
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Dec 10 2006
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