A207136 -id:A207136 - cp's OEIS frontend

A228836 Triangle defined by T(n,k) = binomial(n^2, (n-k)*k), for n>=0, k=0..n, as read by rows.

1, 1, 1, 1, 4, 1, 1, 36, 36, 1, 1, 560, 1820, 560, 1, 1, 12650, 177100, 177100, 12650, 1, 1, 376992, 30260340, 94143280, 30260340, 376992, 1, 1, 13983816, 8217822536, 92263734836, 92263734836, 8217822536, 13983816, 1, 1, 621216192, 3284214703056, 159518999862720, 488526937079580, 159518999862720, 3284214703056, 621216192, 1

Offset: 0

Paul D. Hanna, Sep 05 2013

			The triangle of coefficients C(n^2, (n-k)*k), n>=k, k=0..n, begins:
  1;
  1, 1;
  1, 4, 1;
  1, 36, 36, 1;
  1, 560, 1820, 560, 1;
  1, 12650, 177100, 177100, 12650, 1;
  1, 376992, 30260340, 94143280, 30260340, 376992, 1;
  1, 13983816, 8217822536, 92263734836, 92263734836, 8217822536, 13983816, 1;
  ...

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

Cf. A207136 (row sums), A228837 (antidiagonal sums), A070780 (column 1).

Cf. related triangles: A228900(exp), A209330, A226234, A228832.

T[n_,k_]:=Binomial[n^2, (n-k)*k]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Aug 02 2025 *)

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

A207135 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k)) ).

Original entry on oeis.org

1, 2, 5, 32, 796, 77508, 26058970, 28765221688, 101824384364586, 1145306676113095172, 40618070255705049577152, 4523562146025746408072408406, 1576501611479138389748204925102907, 1714649258669533421310212170714443813118

Offset: 0

Paul D. Hanna, Feb 15 2012

nonn

The logarithmic derivative yields A207136.
Equals the row sums of triangle A228900.
Equals the self-convolution of A228852.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 32*x^3 + 796*x^4 + 77508*x^5 +...
where the logarithm of the g.f. equals the l.g.f. of A207136:
log(A(x)) = 2*x + 6*x^2/2 + 74*x^3/3 + 2942*x^4/4 + 379502*x^5/5 +...

Cf. A207136 (log), A167006, A228900, A206830, A206850, A228852.

{a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m^2,k*(m-k))))+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

A228808 a(n) = Sum_{k=0..n} binomial(n*k, k^2).

Original entry on oeis.org

1, 2, 4, 20, 296, 10067, 927100, 219541877, 110728186648, 137502766579907, 448577320868198789, 3169529341990169816462, 51243646781214826181569316, 2201837465728010770618930322223, 215520476721579201896200887266792583, 45634827026091489574547858030506357191920

Offset: 0

Paul D. Hanna, Sep 04 2013

nonn

Ignoring initial term, equals the logarithmic derivative of A228809.

Equals row sums of triangle A228832.

			L.g.f.: L(x) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 +...
where
exp(L(x)) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 + 158904*x^6 + 31681195*x^7 +...+ A228809(n)*x^n +...

Seiichi Manyama, Table of n, a(n) for n = 0..73

Cf. A228809, A207136, A167009, A228832.

Table[Sum[Binomial[n*k, k^2],{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Sep 06 2013 *)

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

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Sep 06 2013

A238696 a(n) = Sum_{k=0..floor(n/2)} binomial(n(n-k), nk).

Original entry on oeis.org

1, 1, 2, 21, 497, 18508, 3297933, 2348121769, 2319121509374, 4535739243360613, 58887253765506968848, 1694438232474931034462251, 64598311562133275526222276162, 8312693334404799592869803398802772, 5827069387752679429926992257426553147833

Offset: 0

Vaclav Kotesovec, Mar 03 2014

G. C. Greubel, Table of n, a(n) for n = 0..69
Vaclav Kotesovec, Limits, graph for 500 terms

Cf. A206849, A207136, A209331.

Table[Sum[Binomial[n*(n-k), n*k], {k, 0, Floor[n/2]}], {n, 0, 20}]

a(n)=sum(k=0,n\2, binomial(n*(n-k), n*k)) \\ Charles R Greathouse IV, Jul 29 2016

Maximum is at k = n*(1-1/sqrt(5))/2 = 0.2763932... * n.
Limit n->infinity a(n)^(1/n^2) = (1+sqrt(5))/2.
Lim sup n->infinity a(n) / (5^(1/4)/(n*sqrt(2*Pi))*((1+sqrt(5))/2)^(n^2+1)) = JacobiTheta3(0,exp(-5*sqrt(5)/2)) = EllipticTheta[3,0,Exp[-5*Sqrt[5]/2]] = 1.007468786736926147579...
Lim inf n->infinity a(n) / (5^(1/4)/(n*sqrt(2*Pi))*((1+sqrt(5))/2)^(n^2+1)) = JacobiTheta2(0,exp(-5*sqrt(5)/2)) = EllipticTheta[2,0,Exp[-5*Sqrt[5]/2]] = 0.494414344263155315970...
a(n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^(2*n)) for n > 0. - Seiichi Manyama, Oct 11 2021

A207138 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k(n-k))x^k = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 3, 7, 51, 761, 17913, 688745, 56611987, 11405877739, 4272862207703, 2450039810788461, 2224842228379519641, 4169966883810355864393, 19139862395982576668262825, 166161479603614500915921996017, 2206856314384330228779059994929555

Offset: 1

Paul D. Hanna, Feb 15 2012

nonn

Equals the logarithmic derivative of A207137.

			L.g.f.: L(x) =  x + 3*x^2/2 + 7*x^3/3 + 51*x^4/4 + 761*x^5/5 + 17913*x^6/6 +...
where exponentiation equals the g.f. of A207137:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 17*x^4 + 171*x^5 + 3171*x^6 +...
To illustrate the definition, the l.g.f. equals the series:
L(x) = (1 + x)*x + (1 + 4*x + 1*x^2)*x^2/2
+ (1 + 36*x + 36*x^2 + 1*x^3)*x^3/3
+ (1 + 560*x + 1820*x^2 + 560*x^3 + 1*x^4)*x^4/4
+ (1 + 12650*x + 177100*x^2 + 177100*x^3 + 12650*x^4 + 1*x^5)*x^5/5
+ (1 + 376992*x + 30260340*x^2 + 94143280*x^3 + 30260340*x^4 + 376992*x^5 + 1*x^6)*x^6/6 +...

Cf. A207137 (exp), A207136, A167006, A228837.

Table[n*Sum[Binomial[(n-k)^2, k*(n-2*k)]/(n-k),{k,0,Floor[n/2]}],{n,1,20}] (* Vaclav Kotesovec, Mar 04 2014 *)

{a(n)=n*polcoeff(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m^2, k*(m-k))*x^k))+x*O(x^n),n)}

{a(n)=n*sum(k=0,n\2,binomial((n-k)^2,k*(n-2*k))/(n-k))}
for(n=1,20,print1(a(n),", "))

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

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, Mar 04 2014

A207140 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2).

Original entry on oeis.org

1, 2, 10, 407, 56746, 30771252, 115106662819, 1446405270234360, 53819202633553797290, 12313337704248075967333334, 12373818231445938048765251252260, 33156027144321617106970597265032233270, 409476940913917468665022448013012674533441891

Offset: 0

Paul D. Hanna, Feb 15 2012

nonn

Ignoring initial term a(0), equals the logarithmic derivative of A207139.

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 407*x^3/3 + 56746*x^4/4 + 30771252*x^5/5 +...
where exponentiation equals the g.f. of A207139:
exp(L(x)) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
By definition, the initial terms begin: a(0) = 1;
a(1) = C(1,0)*C(1,0), + C(1,1)*C(1,1);
a(2) = C(2,0)*C(4,0), + C(2,1)*C(4,1), + C(2,2)*C(4,4);
a(3) = C(3,0)*C(9,0), + C(3,1)*C(9,1), + C(3,2)*C(9,4), + C(3,3)*C(9,9);
a(4) = C(4,0)*C(16,0), + C(4,1)*C(16,1), + C(4,2)*C(16,4), + C(4,3)*C(16,9), + C(4,4)*C(16,16); ...
which is evaluated as:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 2*4 + 1*1 = 10;
a(3) = 1*1 + 3*9 + 3*126 + 1*1 = 407;
a(4) = 1*1 + 4*16 + 6*1820 + 4*11440 + 1*1 = 56746;
a(5) = 1*1 + 5*25 + 10*12650 + 10*2042975 + 5*2042975 + 1*1 = 30771252;
a(6) = 1*1 + 6*36 + 15*58905 + 20*94143280 + 15*7307872110 + 6*600805296 + 1*1 = 115106662819; ...

Cf. A207139 (exp), A206851, A207136, A207138, A167009.

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

{a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k^2))}
for(n=0,16,print1(a(n),", "))

Limit n->infinity a(n)^(1/n^2) = 2. - Vaclav Kotesovec, Mar 03 2014

A228852 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))/2 ).

Original entry on oeis.org

1, 1, 2, 14, 382, 38344, 12990279, 14369538529, 50897796053428, 572602411324905786, 20308462423438736818782, 2261760763404526386241849803, 788248543938180828988762846368690, 857323841081698966408121705146996762240, 2905542652088907570108828021890682181041282730

Offset: 0

Paul D. Hanna, Sep 05 2013

nonn

Self-convolution yields A207135.

			G.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 382*x^4 + 38344*x^5 + 12990279*x^6 +...
where
log(A(x)) = x + 3*x^2/2 + 37*x^3/3 + 1471*x^4/4 + 189751*x^5/5 + 77708973*x^6/6 +...+ A207136(n)/2 * x^n/n +...

Cf. A207135, A207136.

{a(n)=polcoeff(exp(sum(m=1, n, x^m/m*sum(k=0, m, binomial(m^2, k*(m-k))/2))+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

cp's OEIS Frontend

A228836 Triangle defined by T(n,k) = binomial(n^2, (n-k)*k), for n>=0, k=0..n, as read by rows.

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A207135 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k)) ).

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A228808 a(n) = Sum_{k=0..n} binomial(n*k, k^2).

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A238696 a(n) = Sum_{k=0..floor(n/2)} binomial(n*(n-k), n*k).

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A207138 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))*x^k = Sum_{n>=1} a(n)*x^n/n.

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A207140 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2).

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A228852 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))/2 ).

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A238696 a(n) = Sum_{k=0..floor(n/2)} binomial(n(n-k), nk).

A207138 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k(n-k))x^k = Sum_{n>=1} a(n)*x^n/n.