A206849 -id:A206849 - cp's OEIS frontend

A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).

1, 2, 3, 4, 6, 11, 22, 43, 79, 137, 231, 397, 728, 1444, 3018, 6386, 13278, 26725, 51852, 97243, 177671, 320286, 579371, 1071226, 2053626, 4098627, 8451288, 17742649, 37352435, 77926452, 159899767, 321468048, 632531039, 1219295320, 2308910353, 4314168202

Offset: 0

N. J. A. Sloane, Henry W. Gould

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Henry W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.

Cf. A010052, A206849.

Partial sums of A103198.

[(&+[Binomial(n, j^2): j in [0..n]]): n in [0..50]]; // G. C. Greubel, Oct 26 2022

Table[Sum[Binomial[n, k^2], {k, 0, Sqrt[n]}], {n, 0, 50}] (* T. D. Noe, Sep 10 2011 *)

a(n)=sum(k=0,sqrtint(n),binomial(n,k^2)) \\ Charles R Greathouse IV, Mar 26 2013

def A003099(n): return sum( binomial(n,k^2) for k in range(isqrt(n)+1))
[A003099(n) for n in range(50)] # G. C. Greubel, Oct 26 2022

a(n)*sqrt(n)/2^n is bounded: lim sup a(n)*sqrt(n)/2^n = 0.82... and lim inf a(n)*sqrt(n)/2^n = 0.58... - Benoit Cloitre, Nov 14 2003 [These constants are sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = 0.827112271364145742... and sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = 0.587247586271786487... - Vaclav Kotesovec, Jan 15 2023]
Binomial transform of the characteristic function of squares A010052. - Carl Najafi, Sep 09 2011
G.f.: (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^(k^2). - Ilya Gutkovskiy, Jan 22 2024

More terms from Carl Najafi, Sep 09 2011

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

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

A206848 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) ).

Original entry on oeis.org

1, 2, 5, 53, 3422, 826606, 1335470713, 9548109569885, 190076214495558260, 18558289189760778318731, 10286810587274357297985552184, 16301371794177939084545371104827679, 91249944361047494534207504939405352235731, 3283593155431496336538359592977826684908598341441

Offset: 0

Paul D. Hanna, Feb 15 2012

nonn

Logarithmic derivative yields A206849.

Equals row sums of triangle A228902.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 53*x^3 + 3422*x^4 + 826606*x^5 + 1335470713*x^6 +...
where the logarithm of the g.f. yields the l.g.f. of A206849:
log(A(x)) = 2*x + 6*x^2/2 + 137*x^3/3 + 13278*x^4/4 + 4098627*x^5/5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..57

Cf. A206849 (log), A206846, A206850, A228902.

Cf. variants: A167006, A228809.

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2,k^2))*x^m/m)+x*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))

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(""))

A227403 a(n) = Sum_{k=0..n} binomial(n^2, nk) binomial(n*k, k^2).

Original entry on oeis.org

1, 2, 14, 1514, 1308582, 17304263902, 1362702892177706, 1323407909279927430346, 11218363871234340925730020646, 637467717878006909442727527733810142, 519660435252919757259949810325837093364580014, 2289503386759572781844843312201361014103189493095636611

Offset: 0

Paul D. Hanna, Sep 20 2013

nonn

			The following triangles illustrate the terms involved in the sum
a(n) = Sum_{k=0..n} A209330(n,k) * A228832(n,k).
Triangle A209330(n,k) = binomial(n^2, n*k) 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;
...
Triangle A228832(n,k) = binomial(n*k, k^2) 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; ...

Seiichi Manyama, Table of n, a(n) for n = 0..46
V. Kotesovec, Asymptotic of sequence A227403, Sep 21 2013

Cf. A209330, A228832, A206849, A206152, A237421.

Table[Sum[Binomial[n^2,n*k]*Binomial[n*k,k^2],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 21 2013 *)
r^(-(1+r)^2/(2*r))/.FindRoot[(1-r)^(2*r) == r^(2*r+1), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n^2), Vaclav Kotesovec, Sep 21 2013 *)

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

a(n) = Sum_{k=0..n} (n^2)! / ( (n^2-n*k)! * (n*k-k^2)! * (k^2)! ).

Limit n->infinity a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274..., where r = 0.6032326837741362... (see A237421) is the root of the equation (1-r)^(2*r) = r^(2*r+1). - Vaclav Kotesovec, Sep 21 2013

A206847 a(n) = Sum_{k=0..n} binomial(n^2, k^2) * binomial(n^2, (n-k)^2).

Original entry on oeis.org

1, 2, 18, 2270, 3678482, 51789416252, 9723940840418814, 13783866167176942874214, 260749663122506218247699587346, 35385577627626083328957267246097557212, 64138056102285851525440919122006580387539950268, 814449089808478655249485968539593253265395820497817710866

Offset: 0

Paul D. Hanna, Feb 15 2012

nonn

Ignoring the initial term a(0), equals the logarithmic derivative of A206846.

			L.g.f.: L(x) = 2*x + 18*x^2/2 + 2270*x^3/3 + 3678482*x^4/4 + 51789416252*x^5/5 +...
where exponentiation yields the g.f. of A206846:
exp(L(x)) = 1 + 2*x + 11*x^2 + 776*x^3 + 921193*x^4 + 10359730908*x^5 +...
Illustration of terms: by definition,
a(1) = C(1,0)*C(1,1) + C(1,1)*C(1,0);
a(2) = C(4,0)*C(4,4) + C(4,1)*C(4,1) + C(4,4)*C(4,0);
a(3) = C(9,0)*C(9,9) + C(9,1)*C(9,4) + C(9,4)*C(9,1) + C(9,9)*C(9,0);
a(4) = C(16,0)*C(16,16) + C(16,1)*C(16,9) + C(16,4)*C(16,4) + C(16,9)*C(16,1) + C(16,16)*C(16,0); ...
Numerically, the above evaluates to be:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 4*4 + 1*1 = 18;
a(3) = 1*1 + 9*126 + 126*9 + 1*1 = 2270;
a(4) = 1*1 + 16*11440 + 1820*1820 + 11440*16 + 1*1 = 3678482;
a(5) = 1*1 + 25*2042975 + 12650*2042975 + 2042975*12650 + 2042975*25 + 1*1 = 51789416252; ...

Seiichi Manyama, Table of n, a(n) for n = 0..45

Cf. A206846 (exp), A206849, A206851.

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

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

From Vaclav Kotesovec, Mar 03 2014: (Start)
Limit n->infinity a(n)^(1/n^2) = 16/(3*sqrt(3)).
a(n) ~ c * 2^(4*n^2+3) / (Pi * n^2 * 3^(3*n^2/2+1)), where c = JacobiTheta3(0,9*exp(-16/3)) = EllipticTheta[3, 0, 9*Exp[-16/3]] = 1.08691022925895131... if n is even, and c = JacobiTheta2(0,9*exp(-16/3)) = EllipticTheta[2, 0, 9*Exp[-16/3]] = 0.91485129628884995... if n is odd.
(End)

A066382 a(n) = Sum_{k=0..n} binomial(n^2,k).

Original entry on oeis.org

1, 2, 11, 130, 2517, 68406, 2391496, 102022810, 5130659561, 296881218694, 19415908147836, 1415538531617772, 113796709835547767, 9998149029974754104, 952980844872975079232, 97930011125976327934826, 10791878598088498089377489, 1269466214540655412954317894

Offset: 0

N. J. A. Sloane, Dec 23 2001

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A206849.

[&+[Binomial(n^2, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jun 08 2019

Table[Sum[Binomial[n^2, k], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Jun 08 2019 *)

{ for (n=0, 100, a=0; for (k=0, n, a+=binomial(n^2, k)); write("b066382.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010

a(n) = sum(k=0, n, binomial(n^2,k)); \\ Michel Marcus, Jun 08 2019

a(n) = 2^(n^2) - binomial(n^2, n+1)*hypergeom([1, -n^2+n+1], [2+n], -1). - Vladeta Jovovic, Jul 08 2003

a(n) ~ exp(n - 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Jun 07 2019

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

A306206 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!n!^k).

Original entry on oeis.org

1, 2, 13, 3445, 127028721, 1249195963773451, 5343245431687763366112193, 14729376926426500067331714992293420777, 36332859343341728199556523379140726537646663631786369

Offset: 0

Seiichi Manyama, Jan 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A206849, A227403, A306207.

{a(n) = sum(k=0, n, (n^2)!/((n^2-n*k)!*n!^k))}

From Vaclav Kotesovec, Jan 29 2019: (Start)
a(n) ~ 2 * (n^2)! / (n!)^n.
a(n) ~ n^(n^2 - n/2 + 1) / (exp(1/12) * 2^((n-3)/2) * Pi^((n-1)/2)). (End)

A306207 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!k!^n).

Original entry on oeis.org

1, 2, 19, 9745, 768211081, 17406784944114721, 179762725526880242306609281, 1230064011299573560897489169488350806401, 7660929590740297929124296619236388608530015362840364161

Offset: 0

Seiichi Manyama, Jan 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A206849, A227403, A306206.

{a(n) = sum(k=0, n, (n^2)!/((n^2-n*k)!*k!^n))}

From Vaclav Kotesovec, Jan 29 2019: (Start)
a(n) ~ (n^2)! / (n! * ((n-1)!)^n).
a(n) ~ exp(n - 1/12) * n^(n^2 - n/2 + 1/2) / (2*Pi)^(n/2). (End)

cp's OEIS Frontend

A003099 a(n) = Sum_{k=0..n} binomial(n,k^2).

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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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A206848 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) ).

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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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A227403 a(n) = Sum_{k=0..n} binomial(n^2, n*k) * binomial(n*k, k^2).

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A206847 a(n) = Sum_{k=0..n} binomial(n^2, k^2) * binomial(n^2, (n-k)^2).

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A066382 a(n) = Sum_{k=0..n} binomial(n^2,k).

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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.

A227403 a(n) = Sum_{k=0..n} binomial(n^2, nk) binomial(n*k, k^2).

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

A306206 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!n!^k).

A306207 a(n) = Sum_{k=0..n} (n^2)!/((n^2-nk)!k!^n).