A228832 -id:A228832 - cp's OEIS frontend

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

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(""))

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

Original entry on oeis.org

1, 2, 70, 48620, 601080390, 126410606437752, 442512540276836779204, 25477612258980856902730428600, 23951146041928082866135587776380551750, 365907784099042279561985786395502921046971688680, 90548514656103281165404177077484163874504589675413336841320

Offset: 0

Paul D. Hanna, Dec 02 2011

nonn

Central coefficients of triangle A228832.

			L.g.f.: L(x) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 + ...
where exponentiation equals the g.f. of A201556:
exp(L(x)) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 + ... + A201556(n)*x^n + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..40
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
R. Oblath, Congruences with binomial coefficients, Proceedings of the Indian Academy of Science, Section A, Vol. 1 No. 6, 383-386

Cf. A000984, A007814, A159918, A201556, A214441, A228832, A285388, A285717.

Table[Binomial[2n^2,n^2],{n,0,10}] (* Harvey P. Dale, Dec 10 2011 *)

PARI
```
a(n) = binomial(2*n^2,n^2)
```

from math import comb
def A201555(n): return comb((m:=n**2)<<1,m) # Chai Wah Wu, Jul 08 2022

L.g.f.: ignoring initial term, equals the logarithmic derivative of A201556.
a(n) = (2*n^2)! / (n^2)!^2.
a(n) = Sum_{k=0..n^2} binomial(n^2,k)^2.
For primes p >= 5: a(p) == 2 (mod p^3), Oblath, Corollary II; a(p) == binomial(2*p,p) (mod p^6) - see Mestrovic, Section 5, equation 31. - Peter Bala, Dec 28 2014
A007814(a(n)) = A159918(n). - Antti Karttunen, Apr 27 2017, based on Vladimir Shevelev's Jul 20 2009 formula in A000984.

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

Original entry on oeis.org

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(""))

A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(nk, k^2) y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 26, 62, 1, 1, 5, 70, 1087, 1031, 1, 1, 6, 155, 9257, 124702, 24782, 1, 1, 7, 301, 51397, 4479983, 26375325, 774180, 1, 1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1, 1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1

Offset: 0

Paul D. Hanna, Sep 07 2013

			This triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 7, 1;
1, 4, 26, 62, 1;
1, 5, 70, 1087, 1031, 1;
1, 6, 155, 9257, 124702, 24782, 1;
1, 7, 301, 51397, 4479983, 26375325, 774180, 1;
1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1;
1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1;
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+2*y+y^2)*x^2 + (1+3*y+7*y^2+y^3)*x^3 + (1+4*y+26*y^2+62*y^3+y^4)*x^4 + (1+5*y+70*y^2+1087*y^3+1031*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2*y + y^2)*x^2/2
+ (1 + 3*y + 15*y^2 + y^3)*x^3/3
+ (1 + 4*y + 70*y^2 + 220*y^3 + y^4)*x^4/4
+ (1 + 5*y + 210*y^2 + 5005*y^3 + 4845*y^4 + y^5)*x^5/5
+ (1 + 6*y + 495*y^2 + 48620*y^3 + 735471*y^4 + 142506*y^5 + y^6)*x^6/6 +...
in which the coefficients form A228832(n,k) = binomial(n*k, k^2).

Cf. A228809 (row sums), A228905 (antidiagonal sums), A228906 (diagonal).

Cf. related triangles: A228832 (log), A209196, A228900, A228902.

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

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

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

A228833 a(n) = Sum_{k=0..[n/2]} binomial((n-k)*k, k^2).

Original entry on oeis.org

1, 1, 2, 3, 5, 20, 77, 437, 5509, 54475, 1031232, 31874836, 789351469, 47552777430, 3302430043985, 223753995897916, 39177880844093733, 5954060239110086680, 1226026438114057710320, 551315671593483499670137, 188615011023291125237647365, 124995445742889226418307452940

Offset: 0

Paul D. Hanna, Sep 04 2013

nonn

Equals antidiagonal sums of triangle A228832.

Cf. A228832.

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

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

Limit n->infinity a(n)^(1/n^2) = ((1-r)/(1-2*r))^(r/2) = 1.171233876693210503..., where r = A323773 = 0.366320150305283... is the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r). - Vaclav Kotesovec, Sep 06 2013

A228905 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(nk, k^2) x^k ).

Original entry on oeis.org

1, 1, 2, 3, 5, 12, 33, 139, 1251, 10598, 176642, 4720781, 106779821, 5953841083, 373265833332, 23827795512789, 3914313805097976, 548326897932632059, 108647952177920032693, 45931050219457726501030, 14741338951262398648743248, 9489791738688118291360645939

Offset: 0

Paul D. Hanna, Sep 07 2013

nonn

Equals the antidiagonal sums of triangle A228904.

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 12*x^5 + 33*x^6 + 139*x^7 +...
such that, by definition, the logarithm equals (cf. A228832):
log(A(x)) = (1 + x)*x + (1 + 2*x + x^2)*x^2/2 + (1 + 3*x + 15*x^2 + x^3)*x^3/3 + (1 + 4*x + 70*x^2 + 220*x^3 + x^4)*x^4/4 + (1 + 5*x + 210*x^2 + 5005*x^3 + 4845*x^4 + x^5)*x^5/5 +...
More explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 31*x^5/5 + 114*x^6/6 + 687*x^7/7 + 8679*x^8/8 + 82948*x^9/9 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..120

Cf. A228904, A228832.

Cf. variants: A206850, A207137, A206830.

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

A229052 a(n) = Sum_{k=0..n} binomial(n^2-nk, nk-k^2) * binomial(n*k, k^2).

Original entry on oeis.org

1, 2, 6, 92, 6662, 2150552, 3093730764, 18251332286098, 466740831542894470, 47238803741195397513182, 20522607409110459026633535856, 34700017072200465774261952422246668, 250699892545838622857396499800167790109260, 6984916990466628202550631436961441381064765905022

Offset: 0

Paul D. Hanna, Sep 22 2013

nonn

			The triangle A228832(n,k) = C(n*k, k^2) illustrates the terms involved in the sum a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k):
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; ...

Vincenzo Librandi, Table of n, a(n) for n = 0..50

Cf. A228832, A206847, A218792.

Table[Sum[Binomial[n^2 - n k, n k - k^2] Binomial[n k, k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 22 2013 *)

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

a(n) = Sum_{k=0..n} binomial(n^2-n*k, (n-k)^2) * binomial(n*k, k^2).
a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k).
a(n) = Sum_{k=0..n} (n^2-n*k)! * (n*k)! / ( ((n-k)^2)! * (n*k-k^2)!^2 * (k^2)! ).
a(n) ~ c * 2^(n^2+2)/(Pi*n^2), where c = EllipticTheta[3,0,1/E^2] = 1.271341522189... if n is even and c = EllipticTheta[2,0,1/E^2] = 1.23528676585389... if n is odd. - Vaclav Kotesovec, Sep 22 2013

cp's OEIS Frontend

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

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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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A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * y^k ), as read by rows.

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

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A228833 a(n) = Sum_{k=0..[n/2]} binomial((n-k)*k, k^2).

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A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(nk, k^2) y^k ), as read by rows.

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

A228905 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(nk, k^2) x^k ).

A229052 a(n) = Sum_{k=0..n} binomial(n^2-nk, nk-k^2) * binomial(n*k, k^2).