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A071922
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Unimodal analogue of binomial coefficient, such that A071921(n,m)=a(n+m-1,n) for all (n,m) different from (0,0), arranged in a Pascal-like triangle.
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4
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1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 7, 1, 1, 5, 16, 22, 11, 1, 1, 6, 25, 50, 46, 16, 1, 1, 7, 36, 95, 130, 86, 22, 1, 1, 8, 49, 161, 295, 296, 148, 29, 1, 1, 9, 64, 252, 581, 791, 610, 239, 37, 1, 1, 10, 81, 372, 1036, 1792, 1897, 1163, 367, 46, 1, 1, 11, 100, 525, 1716, 3612
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Also, number of n-length k-ary words avoiding the pattern 1'-2-1". - Ralf Stephan, Apr 28 2004
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LINKS
| S. Kitaev and T. Mansour, Partially ordered generalized patterns and k-ary words.
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FORMULA
| a(n, m)=sum_{k=0}^{n-m} binomial(2k+m-1, 2k).
sum_{m=0}^n a(n, m)=1+F_{2n}, where F_n is the n-th Fibonacci number.
sum_{m=0}^n (-1)^m a(n, m)=1 if 3 divides n, 0 otherwise.
G.f. for k-th row: 1/(1-x)^(2k-1) + sum[j=1..k-1, x/(1-x)^(2j)]. - Ralf Stephan, Apr 28 2004
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MATHEMATICA
| a[n_, m_] := Sum[ Binomial[2k + m - 1, 2k], {k, 0, n - m}]; Flatten[ Table[ a[n, m], {n, 0, 11}, {m, 0, n}]]
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CROSSREFS
| Sequence in context: A122084 A104559 A080853 * A138028 A009999 A144823
Adjacent sequences: A071919 A071920 A071921 * A071923 A071924 A071925
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14, 2002
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EXTENSIONS
| Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 17 2002
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