A228836 -id:A228836 - cp's OEIS frontend

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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, 1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1, 1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1

Offset: 0

Paul D. Hanna, Sep 04 2013

Central coefficients are A201555(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

			The triangle of coefficients C(n*k, k^2), n>=k, k=0..n, begins:
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;
1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1;
1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1; ...

Paul D. Hanna, Rows 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A228808 (row sums), A228833 (antidiagonal sums), A135860 (diagonal), A201555 (central terms).
Cf. A229052.
Cf. related triangles: A228904 (exp), A209330, A226234, A228836.

{T(n, k)=binomial(n*k, k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 84, 84, 1, 1, 1820, 12870, 1820, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1, 1

Offset: 0

Paul D. Hanna, Mar 06 2012

Column 1 equals A014062.
Row sums equal A167009.
Antidiagonal sums equal A209331.
Ignoring initial row T(0,0), equals the logarithmic derivative of the g.f. of triangle A209196.

			The triangle of coefficients C(n^2,n*k), n>=k, k=0..n, begins:
1;
1, 1;
1, 6, 1;
1, 84, 84, 1;
1, 1820, 12870, 1820, 1;
1, 53130, 3268760, 3268760, 53130, 1;
1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1; ...

Paul D. Hanna, Rows n = 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A014062 (column 1), A167009 (row sums), A209331, A209196.
Cf. related triangles: A209196 (exp), A228836, A228832, A226234.
Cf. A206830.

Table[Binomial[n^2, n*k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2018 *)

{T(n,k)=binomial(n^2,n*k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A207136 a(n) = Sum_{k=0..n} binomial(n^2, k*(n-k)).

Original entry on oeis.org

1, 2, 6, 74, 2942, 379502, 155417946, 200991082378, 814134608643518, 10305926982053248142, 406157795399324680023006, 49758289996116571598723737976, 18917910771770463473290738891259546, 22290399373603219140501180230536732389992

Offset: 0

Paul D. Hanna, Feb 15 2012

Ignoring initial term a(0), equals the logarithmic derivative of A207135.

Equals the row sums of triangle A228836.

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 74*x^3/3 + 2942*x^4/4 + 379502*x^5/5 +...
where exponentiation equals the g.f. of A207135:
exp(L(x)) = 1 + 2*x + 5*x^2 + 32*x^3 + 796*x^4 + 77508*x^5 +...
By definition, the initial terms begin: a(0) = 1;
a(1) = C(1,0) + C(1,0);
a(2) = C(4,0) + C(4,1) + C(4,0);
a(3) = C(9,0) + C(9,2) + C(9,2) + C(9,0);
a(4) = C(16,0) + C(16,3) + C(16,4) + C(16,3) + C(16,0);
a(5) = C(25,0) + C(25,4) + C(25,6) + C(25,6) + C(25,4) + C(25,0);
a(6) = C(36,0) + C(36,5) + C(36,8) + C(36,9) + C(36,8) + C(36,5) + C(36,0); ...
which is evaluated as:
a(1) = 1 + 1 = 2;
a(2) = 1 + 4 + 1 = 6;
a(3) = 1 + 36 + 36 + 1 = 74;
a(4) = 1 + 560 + 1820 + 560 + 1 = 2942;
a(5) = 1 + 12650 + 177100 + 177100 + 12650 + 1 = 379502;
a(6) = 1 + 376992 + 30260340 + 94143280 + 30260340 + 376992 + 1 = 155417946; ...

Seiichi Manyama, Table of n, a(n) for n = 0..64

Cf. A207135 (exp), A167009, A228836.

A207136:=n->add(binomial(n^2, k*(n-k)), k=0..n): seq(A207136(n), n=0..15); # Wesley Ivan Hurt, Jun 23 2015

Table[Sum[Binomial[n^2, k*(n-k)],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 03 2014 *)

{a(n)=sum(k=0,n,binomial(n^2,(n-k)*k))}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * 2*sqrt(2/(3*Pi)) * (4/3^(3/4))^(n^2)/n, where c = EllipticTheta[3,0,1/3] = JacobiTheta3(0,1/3) = 1.69145968168171534... if n is even, and c = EllipticTheta[2,0,1/3] = JacobiTheta2(0,1/3) = 1.690611203075214233... if n is odd. - Vaclav Kotesovec, Mar 03 2014

A226234 Triangle defined by T(n,k) = binomial(n^2, k^2), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 126, 1, 1, 16, 1820, 11440, 1, 1, 25, 12650, 2042975, 2042975, 1, 1, 36, 58905, 94143280, 7307872110, 600805296, 1, 1, 49, 211876, 2054455634, 3348108992991, 63205303218876, 262596783764, 1, 1, 64, 635376, 27540584512, 488526937079580, 401038568751465792, 1118770292985239888, 159518999862720, 1

Offset: 0

Paul D. Hanna, Aug 24 2013

Row sums equal A206849.

Antidiagonal sums equal A123165.

			The triangle of coefficients C(n^2,k^2), n>=k, k=0..n, begins:
1;
1, 1;
1, 4, 1;
1, 9, 126, 1;
1, 16, 1820, 11440, 1;
1, 25, 12650, 2042975, 2042975, 1;
1, 36, 58905, 94143280, 7307872110, 600805296, 1;
1, 49, 211876, 2054455634, 3348108992991, 63205303218876, 262596783764, 1;
1, 64, 635376, 27540584512, 488526937079580, 401038568751465792, 1118770292985239888, 159518999862720, 1; ...

Paul D. Hanna, Rows n = 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A206849, A123165.

Cf. related triangles: A228902(exp), A209330, A228832, A228836.

{T(n,k)=binomial(n^2,k^2)}
for(n=0,9,for(k=0,n,print1(T(n,k),", "));print(""))

A228900 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, (n-k)k) y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 15, 15, 1, 1, 155, 484, 155, 1, 1, 2685, 36068, 36068, 2685, 1, 1, 65517, 5082340, 15763254, 5082340, 65517, 1, 1, 2063205, 1179126560, 13201421078, 13201421078, 1179126560, 2063205, 1, 1, 79715229, 411708127954, 19954261054442, 61092286569334, 19954261054442, 411708127954, 79715229, 1

Offset: 0

Paul D. Hanna, Sep 07 2013

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 15, 15, 1;
1, 155, 484, 155, 1;
1, 2685, 36068, 36068, 2685, 1;
1, 65517, 5082340, 15763254, 5082340, 65517, 1;
1, 2063205, 1179126560, 13201421078, 13201421078, 1179126560, 2063205, 1;
1, 79715229, 411708127954, 19954261054442, 61092286569334, 19954261054442, 411708127954, 79715229, 1;
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+15*y+15*y^2+y^3)*x^3 + (1+155*y+484*y^2+155*y^3+y^4)*x^4 + (1+2685*y+36068*y^2+36068*y^3+2685*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 4*y + y^2)*x^2/2
+ (1 + 36*y + 36*y^2 + y^3)*x^3/3
+ (1 + 560*y + 1820*y^2 + 560*y^3 + y^4)*x^4/4
+ (1 + 12650*y + 177100*y^2 + 177100*y^3 + 12650*y^4 + y^5)*x^5/5
+ (1 + 376992*y + 30260340*y^2 + 94143280*y^3 + 30260340*y^4 + 376992*y^5 + y^6)*x^6/6 +...
in which the coefficients form A228836(n,k) = binomial(n^2, (n-k)*k).

Cf. A207135 (row sums), A207137 (antidiagonal sums), A228901 (column 1).

Cf. related triangles: A228836 (log), A209196, A228902, A228904.

{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m^2, (m-j)*j)*y^j))+x*O(x^n)), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2k)k).

Original entry on oeis.org

1, 1, 2, 5, 38, 597, 14472, 554653, 44421258, 8933194659, 3408672951784, 1984802013951149, 1803179670478111304, 3323206887194925488269, 15156709454119350064982141, 132889643918499982093215167857, 1784438297905511051093397284187186

Offset: 0

Paul D. Hanna, Sep 05 2013

nonn

Equals the antidiagonal sums of triangle A228836.

G. C. Greubel, Table of n, a(n) for n = 0..86

Cf. A228836, A207138.

Cf. variants: A209331, A228833, A123165.

Table[Sum[Binomial[(n-k)^2, (n-2*k)*k],{k,0,Floor[n/2]}],{n,0,15}] (* Vaclav Kotesovec, Sep 05 2013 *)

{a(n)=sum(k=0,n\2,binomial((n-k)^2, (n-2*k)*k))}
for(n=0,30,print1(a(n),", "))

Limit n->infinity a(n)^(1/n^2) = ((1-r)^2/(r*(1-2*r)))^((1-3*r)*(1-r)/(3*(1-2*r))) = 1.36198508972775011599..., where r = 0.195220321930105755... is the root of the equation (1-3*r+3*r^2)^(3*(2*r-1)) = (r*(1-2*r))^(4*r-1) * (1-r)^(4*(r-1)). - Vaclav Kotesovec, added Sep 05 2013, simplified Mar 04 2014

cp's OEIS Frontend

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.

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A207136 a(n) = Sum_{k=0..n} binomial(n^2, k*(n-k)).

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A226234 Triangle defined by T(n,k) = binomial(n^2, k^2), for n>=0, k=0..n, as read by rows.

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A228900 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, (n-k)*k) * y^k ), as read by rows.

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A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2*k)*k).

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A228900 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, (n-k)k) y^k ), as read by rows.

A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2k)k).