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Search: id:A008459
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| A008459 |
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Square the entries of Pascal's triangle. |
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+0
31
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| 1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 36, 16, 1, 1, 25, 100, 100, 25, 1, 1, 36, 225, 400, 225, 36, 1, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 64, 784, 3136, 4900, 3136, 784, 64, 1, 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1, 1, 100, 2025, 14400, 44100, 63504
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Number of lattice paths from (0,0) to (n,n) with steps (1,0) and (0,1), having k right turns. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 23 2003
Product of A007318 and A105868. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2005
Number of partitions that fit in an n X n box with Durfee square k. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 20 2006
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REFERENCES
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J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650
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LINKS
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A. Necer, Series formelles et produit de Hadamard
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FORMULA
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Cf. A007318, A055133.
E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 17 2003
G.f.: 1/sqrt(1-2*y-2*x*y+y^2-2*x*y^2+x^2*y^2); g.f. for row n: (1-t)^n P_n[(1+t)/(1-t)] where the P_n's are the Legendre polynomials. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 23 2003
G.f. for column k is sum(C(k, j)^2*x^(k+j), j, 0, k)/(1-x)^(2k+1). - Paul Barry (pbarry(AT)wit.ie), Nov 15 2005
Column k has g.f. x^k*Legendre_P(k, (1+x)/(1-x))/(1-x)^(k+1)=x^k*sum{j=0..k, C(k, j)^2*x^j}/(1-x)^(2k+1). - Paul Barry (pbarry(AT)wit.ie), Nov 19 2005
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EXAMPLE
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1; 1,1; 1,4,1; 1,9,9,1; 1,16,36,16,1; ...
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MAPLE
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binomial(n, k)^2;
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PROGRAM
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(PARI) T(n, k)=if(k<0|k>n, 0, binomial(n, k)^2)
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CROSSREFS
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Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
Cf. A116647.
Sequence in context: A060102 A082043 A124216 this_sequence A039756 A126065 A126062
Adjacent sequences: A008456 A008457 A008458 this_sequence A008460 A008461 A008462
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KEYWORD
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nonn,tabl
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AUTHOR
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njas
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