A090642
Triangle read by rows: T(n,k) = binomial(n^2, k), 0 <= k <= n.
Original entry on oeis.org
1, 1, 1, 1, 4, 6, 1, 9, 36, 84, 1, 16, 120, 560, 1820, 1, 25, 300, 2300, 12650, 53130, 1, 36, 630, 7140, 58905, 376992, 1947792, 1, 49, 1176, 18424, 211876, 1906884, 13983816, 85900584, 1, 64, 2016, 41664, 635376, 7624512, 74974368, 621216192, 4426165368
Offset: 0
Triangle begins:
1;
1, 1;
1, 4, 6;
1, 9, 36, 84;
1, 16, 120, 560, 1820;
1, 25, 300, 2300, 12650, 53130;
1, 36, 630, 7140, 58905, 376992, 1947792;
...
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), ", "))
A007819
a(n) = Sum_{j=1..n} binomial(n^2, j).
Original entry on oeis.org
1, 10, 129, 2516, 68405, 2391495, 102022809, 5130659560, 296881218693, 19415908147835, 1415538531617771, 113796709835547766, 9998149029974754103, 952980844872975079231, 97930011125976327934825
Offset: 1
Joseph Lavinus Ganley (jwl8k(AT)server.cs.Virginia.EDU)
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[ &+[Binomial(n^2,j): j in [1..n]]: n in [1..20]]; // G. C. Greubel, Mar 06 2020
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seq( add(binomial(n^2,j), j=1..n), n=1..20); # G. C. Greubel, Mar 06 2020
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Table[Sum[Binomial[n^2,i],{i,n}],{n,20}] (* Harvey P. Dale, Nov 03 2013 *)
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vector(20, n, sum(j=1,n, binomial(n^2,j))) \\ G. C. Greubel, Mar 06 2020
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[sum(binomial(n^2,j) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Mar 06 2020
Showing 1-3 of 3 results.
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