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A065942
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Central column of triangle A065941.
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5
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1, 1, 3, 4, 15, 21, 84, 120, 495, 715, 3003, 4368, 18564, 27132, 116280, 170544, 735471, 1081575, 4686825, 6906900, 30045015, 44352165, 193536720, 286097760, 1251677700, 1852482996, 8122425444, 12033222880, 52860229080, 78378960360
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OFFSET
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0,3
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COMMENTS
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When viewed as (1,1), (3,4), (15,21), ... this represents a shallow staircase on Pascal's triangle, arranged as a square array. - Paul Barry, Mar 11 2003
Also central column of triangle A011973 (taking rows with odd number of terms only). - John Molokach, Jul 08 2013
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REFERENCES
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Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)
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LINKS
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FORMULA
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a(n) = binomial(2n-floor((n+1)/2), floor(n/2)).
a(n+1) = Sum_{k=0..ceiling(n/2)} binomial(n+k, k). - Benoit Cloitre, Mar 06 2004
a(n) = binomial(n+floor(n/2), n). - Paul Barry, May 18 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+k, k). - Paul Barry, Jul 06 2004
a(2n-1) = binomial(3n-3,n-1); a(2n) = binomial(3n-2,n-1). - John Molokach, Jul 08 2013
G.f.: A(x) = x*(d/dx)[log(S(x)-1)] = x*[(d/dx) S(x)]/[S(x)-1], where S(x) is the g.f. of A047749. - Vladimir Kruchinin, Jun 12 2014.
Conjecture: 8*n*(n-1)*a(n) -36*(n-1)*(n-3)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Jun 13 2014
0 = a(n)*(+281138850*a(n+2) +729089100*a(n+3) -77071527*a(n+4) -134472793*a(n+5)) +a(n+1)*(+15618825*a(n+2) -1650969*a(n+3) -9342280*a(n+4) -1729448*a(n+5)) +a(n+2)*(-19089675*a(n+2) -61394833*a(n+3) +6470716*a(n+4) +14929796*a(n+5)) +a(n+3)*(-1291668*a(n+3) +553572*a(n+4) +246032*a(n+5)) for all n in Z. - Michael Somos, Jun 23 2018
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EXAMPLE
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G.f. = 1 + x + 3*x^2 + 4*x^3 + 15*x^4 + 21*x^5 + 84*x^6 + 120*x^7 + ... - Michael Somos, Jun 23 2018
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MATHEMATICA
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PROG
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(PARI) a(n) = binomial(n+n\2, n); \\ Altug Alkan, Apr 24 2018
(GAP) List([0..40], n->Binomial(n+Int(n/2), n)); # Muniru A Asiru, Apr 28 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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