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Search: id:A125812
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| A125812 |
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q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2. |
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+0
7
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| 1, 1, 2, 6, 28, 204, 2344, 43160, 1291952, 63647664, 5218320672, 719221578080, 168115994031040, 67159892835119296, 46166133463916209792, 54941957091151982047616, 113826217192695041078973184
(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} A022166(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} A022166(n-1,k) * a(k)
is illustrated by:
a(2) = 1*(1) + 3*(1) + 1*(2) = 6;
a(3) = 1*(1) + 7*(1) + 7*(2) + 1*(6) = 28;
a(4) = 1*(1) + 15*(1) + 35*(2) + 15*(6) + 1*(28) = 204.
Triangle A022166 begins:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 35, 15, 1;
1, 31, 155, 155, 31, 1;
1, 63, 651, 1395, 651, 63, 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=2: */ {a(n)=subst(B_q(n), q, 2)}
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CROSSREFS
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Cf. A022166; A125810, A125811, A125813, A125814, A125815.
Sequence in context: A111342 A008964 A058128 this_sequence A093657 A006117 A118025
Adjacent sequences: A125809 A125810 A125811 this_sequence A125813 A125814 A125815
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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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