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A005191
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Central pentanomial coefficients: largest coefficient of (1+x+...+x^4)^n.
(Formerly M3891)
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47
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1, 1, 5, 19, 85, 381, 1751, 8135, 38165, 180325, 856945, 4091495, 19611175, 94309099, 454805755, 2198649549, 10651488789, 51698642405, 251345549849, 1223798004815, 5966636799745, 29125608152345, 142330448514875, 696235630761115
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Coefficient of x^n in ((1-x^10)/((1-x^5)(1-x^2)(1-x)))^n. - Michael Somos, Sep 24 2003
Note that n divides a(n+1)-a(n). - T. D. Noe (noe(AT)sspectra.com), Mar 16 2005
Terms that are not a multiple of 5 have zero density, namely, there are fewer than n^(log(4)/log(5)) such terms among A005191(1..n). In particular, A005191(5k+2) and A005191(5k+4) are multiples of 5 for every k. - Max Alekseyev (maxale(AT)gmail.com), Apr 25 2005
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REFERENCES
| V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
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FORMULA
| a(n) = sum(k=0..[2n/5], binomial(n, k)*binomial(-n, 2n-5k) ); a(n) = (5^n + sum(j=1..2n-1, (sin(5j*Pi/(2n))/sin(j*Pi/(2n)))^n))/(2n) - 2. - Max Alekseyev (maxale(AT)gmail.com), Mar 04 2005
Conjecture: 2*n*(2*n-1)*(3*n-4)*a(n) -(3*n-1)*(19*n^2-38*n+18)*a(n-1) -5*(n-1)*(3*n-4)*(2*n-1)*a(n-2) +25*(n-1)*(n-2)*(3*n-1)*a(n-3) = 0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 21 2010]
Conjecture : a(5*n+2) and a(5*n+4) are congruent to 0 mod 5. [From Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 07 2010]
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(((1-x^5)/(1-x)+x*O(x^(2*n)))^n, 2*n))
(PARI) a(n)=if(n<0, 0, polcoeff(((1-x^10)/((1-x^5)*(1-x^2)*(1-x))+x*O(x^n))^n, n))
(PARI) a(n) = sum(k=0, (2*n)\5, binomial(n, k)*binomial(-n, 2*n-5*k)) a(n) = round((5^n+sum(j=1, 2*n-1, (sin(5*Pi*j/2/n)/sin(Pi*j/2/n))^n))/2/n)-2 (Alekseyev)
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CROSSREFS
| Cf. A001405, A002426, A005190, A018901, A025012, A025013, A025014
Sequence in context: A149794 A149795 A149796 * A147091 A149797 A149798
Adjacent sequences: A005188 A005189 A005190 * A005192 A005193 A005194
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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