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A008459 Square the entries of Pascal's triangle. +0
31
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)
OFFSET

0,5

COMMENT

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

REFERENCES

J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650

LINKS

A. Necer, Series formelles et produit de Hadamard

FORMULA

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

EXAMPLE

1; 1,1; 1,4,1; 1,9,9,1; 1,16,36,16,1; ...

MAPLE

binomial(n, k)^2;

PROGRAM

(PARI) T(n, k)=if(k<0|k>n, 0, binomial(n, k)^2)

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

njas

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Last modified July 25 07:41 EDT 2008. Contains 142293 sequences.


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