A123165 -id:A123165 - cp's OEIS frontend

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.

1, 3, 7, 15, 231, 2763, 37773, 3347359, 145164760, 15115517783, 5300285945494, 841490209145991, 700215432847179640, 821522962294608211319, 580955012898669141073842, 3240132942509582109732641935, 12114306457535986210506222037102

Offset: 1

Paul D. Hanna, Feb 15 2012

nonn

Equals the logarithmic derivative of A206850.

			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 +...

Seiichi Manyama, Table of n, a(n) for n = 1..94

Cf. A206850 (exp), A206847, A206849, A123165.

Table[n*Sum[Binomial[(n-k)^2, k^2]/(n-k),{k,0,Floor[n/2]}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

{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)}

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

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

Limit n->infinity a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166349..., where r = A323778 = 0.365498498219858044579... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - 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(""))

A323778 Decimal expansion of the root of the equation (1-r)^(2-2r) r^(2r) = 1-2r.

Original entry on oeis.org

3, 6, 5, 4, 9, 8, 4, 9, 8, 2, 1, 9, 8, 5, 8, 0, 4, 4, 5, 7, 9, 7, 3, 6, 8, 7, 5, 4, 4, 6, 2, 9, 9, 0, 8, 8, 3, 2, 2, 7, 5, 8, 8, 0, 6, 9, 6, 3, 4, 6, 0, 2, 9, 5, 0, 1, 5, 9, 5, 5, 1, 6, 7, 6, 8, 2, 1, 1, 8, 8, 3, 6, 7, 4, 0, 8, 4, 8, 7, 3, 0, 0, 3, 5, 2, 2, 8, 4, 1, 0, 7, 4, 0, 8, 2, 1, 5, 4, 8, 5, 3, 8, 7, 5, 7, 8

Offset: 0

Vaclav Kotesovec, Jan 27 2019

			0.36549849821985804457973687544629908832275880696346029501595516768211883674...

Cf. A123165, A206851.

Cf. A220359, A237421, A245259, A323768, A323773, A323777.

RealDigits[r/.FindRoot[(1-r)^(2-2*r) * r^(2*r) == 1-2*r, {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

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

A123163 Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 0, 0, 1, 9, 1, 0, 0, 1, 16, 126, 0, 0, 0, 1, 25, 1820, 1, 0, 0, 0, 1, 36, 12650, 11440, 0, 0, 0, 0, 1, 49, 58905, 2042975, 1, 0, 0, 0, 0, 1, 64, 211876, 94143280, 2042975, 0, 0, 0, 0, 0, 1, 81, 635376, 2054455634, 7307872110, 1, 0, 0, 0, 0, 0

Offset: 0

Roger L. Bagula, Oct 02 2006

			n\k | 0    1      2      3    4    5    6    7
----+--------------------------------------------
  0 | 1;
  1 | 1,   0;
  2 | 1,   1,     0;
  3 | 1,   4,     0,     0;
  4 | 1,   9,     1,     0,   0;
  5 | 1,  16,   126,     0,   0,   0;
  6 | 1,  25,  1820,     1,   0,   0,   0;
  7 | 1,  36, 12650, 11440,   0,   0,   0,   0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A011973, A123165.

[Binomial((n-k)^2, k^2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023

T[n_, k_]= (n^2-2*n*k+k^2)!/((k^2)!(n^2-2*n*k)!);
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten
Flatten[Table[Binomial[(n-m)^2,m^2],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Aug 08 2012 *)

flatten([[binomial((n-k)^2, k^2) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 18 2023

T(n, k) = (n^2 - 2*n*k + k^2)!/((k^2)!(n^2 - 2*n*k)!).
From G. C. Greubel, Jul 18 2023: (Start)
T(n, 0) = T(2*n, n) = 1.
T(n, n) = A000007(n).
Sum_{k=0..n} T(n, k) = A123165(n). (End)

Edited by N. J. A. Sloane, Oct 04 2006

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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.

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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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A323778 Decimal expansion of the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r.

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

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A123163 Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows.

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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.

A323778 Decimal expansion of the root of the equation (1-r)^(2-2r) r^(2r) = 1-2r.

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