A137635
a(n) = Sum_{k=0..n} C(2k,k)*C(2k,n-k); equals row 0 of square array A137634.
Original entry on oeis.org
1, 2, 10, 46, 226, 1136, 5810, 30080, 157162, 826992, 4376408, 23267332, 124179570, 664919780, 3570265000, 19216805476, 103652442922, 560127574340, 3031887311256, 16435458039076, 89213101943000, 484839755040768, 2637805800869740, 14365506336197816
Offset: 0
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CoefficientList[Series[1/Sqrt[1 - 4*x*(1 + x)^2],{x,0,50}],x] (* Stefano Spezia, Sep 01 2018 *)
Table[Sum[Binomial[2k,k]Binomial[2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Dec 31 2018 *)
a[n_]:=Binomial[2n, n]HypergeometricPFQ[{(1-2*n)/3, 2(1-n)/3, -2n/3}, {1/2-n, 1/2-n}, -3^3/2^4]; Array[a,24,0] (* Stefano Spezia, Jul 11 2024 *)
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a(n)=sum(k=0,n,binomial(2*k,k)*binomial(2*k,n-k));
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a(n)=polcoeff(1/sqrt(1-4*x*(1+x +x*O(x^n))^2),n,x); /* Using the g.f.: */
A137637
a(n) = Sum_{k=0..n} C(2k+2,k)*C(2k+2,n-k) ; equals row 2 of square array A137634 ; also equals the convolution of A137635 and the self-convolution of A073157.
Original entry on oeis.org
1, 6, 32, 170, 899, 4764, 25318, 134964, 721562, 3868024, 20785035, 111931154, 603938905, 3264309644, 17671408012, 95800342628, 520022296700, 2826089180652, 15374990077568, 83727902852188, 456370687687082
Offset: 0
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{a(n)=sum(k=0,n,binomial(2*k+2,k)*binomial(2*k+2,n-k))} /* Using the g.f.: */ {a(n)=local(R=1/sqrt(1-4*x*(1+x +x*O(x^n))^2), G=(1-sqrt(1-4*x*(1+x)^2+x^2*O(x^n)))/(2*x*(1+x+x*O(x^n)))); polcoeff(R*G^2,n,x)}
A137634
Square array where T(n,k) = Sum_{j=0..k} C(n+2*j,j)*C(n+2*j,k-j), read by antidiagonals.
Original entry on oeis.org
1, 1, 2, 1, 4, 10, 1, 6, 19, 46, 1, 8, 32, 94, 226, 1, 10, 49, 170, 474, 1136, 1, 12, 70, 282, 899, 2431, 5810, 1, 14, 95, 438, 1577, 4764, 12609, 30080, 1, 16, 124, 646, 2600, 8701, 25318, 65972, 157162, 1, 18, 157, 914, 4076, 15000, 47682, 134964, 347524, 826992
Offset: 0
Square array begins:
1, 2, 10, 46, 226, 1136, 5810, 30080, 157162, ...;
1, 4, 19, 94, 474, 2431, 12609, 65972, 347524, ...;
1, 6, 32, 170, 899, 4764, 25318, 134964, 721562, ...;
1, 8, 49, 282, 1577, 8701, 47682, 260384, 1419436, ...;
1, 10, 70, 438, 2600, 15000, 85102, 477808, 2664539, ...;
1, 12, 95, 646, 4076, 24643, 145099, 839620, 4800849, ...;
1, 14, 124, 914, 6129, 38868, 237842, 1420660, 8342297, ...;
1, 16, 157, 1250, 8899, 59201, 376740, 2325088, 14036647, ...; ...
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{T(n,k)=sum(j=0,k,binomial(n+2*j,j)*binomial(n+2*j,k-j))} /* Using the g.f.: */ {T(n,k)=local(Oy=y*O(y^(n+k))); polcoeff(polcoeff(1/sqrt(1-4*y*(1+y)^2+Oy)* 1/(1-x*((1-sqrt(1-4*y*(1+y)^2+Oy))/(2*y*(1 + y+Oy))+x*O(x^n))),n,x),k,y)}
A137638
Antidiagonal sums of square array A137634.
Original entry on oeis.org
1, 3, 15, 72, 361, 1840, 9505, 49578, 260540, 1377328, 7316373, 39020372, 208809544, 1120621368, 6029023185, 32507001876, 175604614108, 950233307930, 5149691511432, 27946158749572, 151843410356906, 825949622559366
Offset: 0
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{a(n)=sum(k=0,n,sum(j=0,k,binomial(2*j+n-k,j)*binomial(2*j+n-k,k-j)))} /* Using the g.f.: */ {a(n)=local(G=sqrt(1 - 4*x*(1+x)^2 +x*O(x^n))); polcoeff(2*(1+x)/((1+2*x+G)*G),n)}
Showing 1-4 of 4 results.
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