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Search: id:A125815
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| A125815 |
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q-Bell numbers for q=5; eigensequence of A022169, which is the triangle of Gaussian binomial coefficients [n,k] for q=5. |
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+0
6
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| 1, 1, 2, 9, 103, 3276, 307867, 89520089, 83657942588, 258923776689771, 2717711483011792407, 98702105953049319472394, 12629828399521800714941435773, 5784963467206342855747483263957541
(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} A022169(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} A022169(n-1,k) * a(k)
is illustrated by:
a(2) = 1*(1) + 6*(1) + 1*(2) = 9;
a(3) = 1*(1) + 31*(1) + 31*(2) + 1*(9) = 103;
a(4) = 1*(1) + 156*(1) + 806*(2) + 156*(9) + 1*(103) = 3276.
Triangle A022169 begins:
1;
1, 1;
1, 6, 1;
1, 31, 31, 1;
1, 156, 806, 156, 1;
1, 781, 20306, 20306, 781, 1;
1, 3906, 508431, 2558556, 508431, 3906, 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=5: */ {a(n)=subst(B_q(n), q, 5)}
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CROSSREFS
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Cf. A022169; A125810, A125811, A125812, A125813, A125814.
Sequence in context: A013520 A041239 A098610 this_sequence A132494 A136172 A012986
Adjacent sequences: A125812 A125813 A125814 this_sequence A125816 A125817 A125818
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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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