A206849
a(n) = Sum_{k=0..n} binomial(n^2, k^2).
Original entry on oeis.org
1, 2, 6, 137, 13278, 4098627, 8002879629, 66818063663192, 1520456935214867934, 167021181249536494996841, 102867734705055054467692090431, 179314863425920182637610314008444247, 1094998941099523423274757578750950802034789
Offset: 0
L.g.f.: L(x) = 2*x + 6*x^2/2 + 137*x^3/3 + 13278*x^4/4 + 4098627*x^5/5 +...
where exponentiation yields the g.f. of A206848:
exp(L(x)) = 1 + 2*x + 5*x^2 + 53*x^3 + 3422*x^4 + 826606*x^5 + 1335470713*x^6 +...
Illustration of terms: by definition,
a(1) = C(1,0) + C(1,1);
a(2) = C(4,0) + C(4,1) + C(4,4);
a(3) = C(9,0) + C(9,1) + C(9,4) + C(9,9);
a(4) = C(16,0) + C(16,1) + C(16,4) + C(16,9) + C(16,16);
a(5) = C(25,0) + C(25,1) + C(25,4) + C(25,9) + C(25,16) + C(25,25);
a(6) = C(36,0) + C(36,1) + C(36,4) + C(36,9) + C(36,16) + C(36,25) + C(36,36); ...
Numerically, the above evaluates to be:
a(1) = 1 + 1 = 2;
a(2) = 1 + 4 + 1 = 6;
a(3) = 1 + 9 + 126 + 1 = 137;
a(4) = 1 + 16 + 1820 + 11440 + 1 = 13278;
a(5) = 1 + 25 + 12650 + 2042975 + 2042975 + 1 = 4098627;
a(6) = 1 + 36 + 58905 + 94143280 + 7307872110 + 600805296 + 1 = 8002879629; ...
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Table[Sum[Binomial[n^2, k^2],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 03 2014 *)
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{a(n)=sum(k=0, n,binomial(n^2,k^2))}
for(n=0, 20, print1(a(n), ", "))
A206851
L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)*x^k = Sum_{n>=1} a(n)*x^n/n.
Original entry on oeis.org
1, 3, 7, 15, 231, 2763, 37773, 3347359, 145164760, 15115517783, 5300285945494, 841490209145991, 700215432847179640, 821522962294608211319, 580955012898669141073842, 3240132942509582109732641935, 12114306457535986210506222037102
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 231*x^5/5 + 2763*x^6/6 +...
where exponentiation yields the g.f. of A206850:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...
By definition, the l.g.f. equals the series:
L(x) = (C(1,0) + C(1,1)*x)*x
+ (C(4,0) + C(4,1)*x + C(4,4)*x^2)*x^2/2
+ (C(9,0) + C(9,1)*x + C(9,4)*x^2 + C(9,9)*x^3)*x^3/3
+ (C(16,0) + C(16,1)*x + C(16,4)*x^2 + C(16,9)*x^3 + C(16,16)*x^4)*x^4/4
+ (C(25,0) + C(25,1)*x + C(25,4)*x^2 + C(25,9)*x^3 + C(25,16)*x^4 + C(25,25)*x^5)*x^5/5 +...
More explicitly,
L(x) = (1 + 1*x)*x + (1 + 4*x + 1*x^2)*x^2/2
+ (1 + 9*x + 126*x^2 + 1*x^3)*x^3/3
+ (1 + 16*x + 1820*x^2 + 11440*x^3 + 1*x^4)*x^4/4
+ (1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + 1*x^5)*x^5/5
+ (1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + 1*x^6)*x^6/6 +...
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Table[n*Sum[Binomial[(n-k)^2, k^2]/(n-k),{k,0,Floor[n/2]}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
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{a(n)=n*polcoeff(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m^2,k^2)*x^k)+x*O(x^n)),n)}
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{a(n)=n*sum(k=0,n\2, binomial((n-k)^2, k^2)/(n-k))}
for(n=1, 20, print1(a(n), ", "))
A229052
a(n) = Sum_{k=0..n} binomial(n^2-n*k, n*k-k^2) * binomial(n*k, k^2).
Original entry on oeis.org
1, 2, 6, 92, 6662, 2150552, 3093730764, 18251332286098, 466740831542894470, 47238803741195397513182, 20522607409110459026633535856, 34700017072200465774261952422246668, 250699892545838622857396499800167790109260, 6984916990466628202550631436961441381064765905022
Offset: 0
The triangle A228832(n,k) = C(n*k, k^2) illustrates the terms involved in the sum a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k):
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1; ...
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Table[Sum[Binomial[n^2 - n k, n k - k^2] Binomial[n k, k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 22 2013 *)
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{a(n)=sum(k=0,n,binomial(n^2-n*k,n*k-k^2)*binomial(n*k,k^2))}
for(n=0,20,print1(a(n),", "))
A206846
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2,k^2) * binomial(n^2,(n-k)^2) ).
Original entry on oeis.org
1, 2, 11, 776, 921193, 10359730908, 1620677532919905, 1969126979596399128130, 32593711828578589304123599877, 3931730912701446701027876250509820962, 6413805618092047206104426809813307222469463650, 74040826359052943559114050244071546075856822107307951070
Offset: 0
G.f.: A(x) = 1 + 2*x + 11*x^2 + 776*x^3 + 921193*x^4 + 10359730908*x^5 +...
where the logarithm of the g.f. yields the l.g.f. of A206847:
log(A(x)) = 2*x + 18*x^2/2 + 2270*x^3/3 + 3678482*x^4/4 + 51789416252*x^5/5 +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2,k^2)*binomial(m^2,(m-k)^2))*x^m/m)+x*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))
Showing 1-4 of 4 results.
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