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A005725
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Quadrinomial coefficients.
(Formerly M2843)
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3
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1, 1, 3, 10, 31, 101, 336, 1128, 3823, 13051, 44803, 154518, 534964, 1858156, 6472168, 22597760, 79067375, 277164295, 973184313, 3422117190, 12049586631, 42478745781, 149915252028, 529606271560, 1872653175556, 6627147599476
(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+x^2+x^3)^n.
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), (1,3). [Joerg Arndt, Jul 05 2011]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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(i+j+k=n, 0<=k<=j<=i<=n, binomial(n, i)*binomial(i, j)*binomial(j, k)) - Benoit Cloitre, Jun 06 2004
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MAPLE
| seq( sum('binomial(n, 2*k)*binomial(n, k)', 'k'=0..floor(n/2)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
a:=n->sum(binomial(n, 1*j)*binomial(n, 2*j), j=0..n): seq(a(n), n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 12 2007
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MATHEMATICA
| a[n_] := Coefficient[(1+x+x^2+x^3)^n, x^n]; a[0] = 1; Table[a[n], {n, 0, 25}] (* From Jean-François Alcover, Nov 15 2011 *)
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PROG
| (Maxima) quadrinomial(n, k):=coeff(expand((1+x+x^2+x^3)^n), x, k);
makelist(quadrinomial(n, n), n, 0, 12); [Emanuele Munarini, Mar 15 2011]
(MAGMA) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+x^2+x^3)^n)[n+1]: n in [0..25] ]; // Bruno Berselli, Jul 05 2011
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CROSSREFS
| Sequence in context: A114487 A017934 A005510 * A079522 A024426 A034016
Adjacent sequences: A005722 A005723 A005724 * A005726 A005727 A005728
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 12 2000
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