A209330 -id:A209330 - cp's OEIS frontend

A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

1, 2, 8, 170, 16512, 6643782, 11582386286, 79450506979090, 2334899414608412672, 265166261617029717011822, 128442558588779813655233443038, 238431997806538515396060130910954852

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			The triangle A209330 of coefficients C(n^2, n*k), n>=k>=0, 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; ...
in which the row sums form this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..58
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A014062, A167006, A167010, A209330, A218792, A306846.

[(&+[Binomial(n^2, n*j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Table[Sum[Binomial[n^2,n*k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Dec 11 2011 *)

PARI
```
a(n)=sum(k=0,n,binomial(n^2,n*k))
```

[sum(binomial(n^2, n*j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Ignoring initial term, equals the logarithmic derivative of A167006. - Paul D. Hanna, Nov 18 2009
If n is even then a(n) ~ c * 2^(n^2 + 1/2)/(n*sqrt(Pi)), where c = Sum_{k = -infinity..infinity} exp(-2*k^2) = 1.271341522189... (see A218792). - Vaclav Kotesovec, Nov 05 2012
If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - Vaclav Kotesovec, Nov 06 2012
a(n) = A306846(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^n) for n > 0. - Seiichi Manyama, Oct 11 2021

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

Original entry on oeis.org

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(""))

A206830 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, nk) x^k ).

Original entry on oeis.org

1, 1, 2, 5, 34, 520, 14397, 993806, 222547738, 98753510701, 66772601607218, 82150206439975648, 310163020349941301606, 3022167582612808506550780, 47176617497043375266215814522, 1129578055293824008530028604347686, 62478430488069985838347598494293429802

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Note: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n^2} binomial(n^2, k) * x^k ) does not yield an integer series (see A227467).

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 34*x^4 + 520*x^5 + 14397*x^6 + ...
such that, by definition, the logarithm equals:
log(A(x)) = x*(1+x) + x^2*(1 + 6*x + x^2)/2 + x^3*(1 + 84*x + 84*x^2 + x^3)/3 + x^4*(1 + 1820*x + 12870*x^2 + 1820*x^3 + x^4)/4 + x^5*(1 + 53130*x + 3268760*x^2 + 3268760*x^3 + 53130*x^4 + x^5)/5 + ... + x^n/n*Sum_{k=0..n} A209330(n,k)*x^k + ...
More explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 115*x^4/4 + 2416*x^5/5 + 83064*x^6/6 + ...

Cf. A167006, A201556, A227467, A209330, A207137, A228905.

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2, m*k)*x^k)*x^m/m)+x*O(x^n)), n)}
for(n=0,15,print1(a(n),", "))

A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, nk) y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 32, 32, 1, 1, 487, 3282, 487, 1, 1, 11113, 657573, 657573, 11113, 1, 1, 335745, 209282906, 1513844855, 209282906, 335745, 1, 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1, 1, 565877928, 61162554558200

Offset: 0

Paul D. Hanna, Mar 05 2012

			This triangle begins:
1;
1, 1;
1, 4, 1;
1, 32, 32, 1;
1, 487, 3282, 487, 1;
1, 11113, 657573, 657573, 11113, 1;
1, 335745, 209282906, 1513844855, 209282906, 335745, 1;
1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1;
1, 565877928, 61162554558200, 31336815578461815, 229089181252258800, 31336815578461815, 61162554558200, 565877928, 1; ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+4*y+y^2)*x^2 + (1+32*y+32*y^2+y^3)*x^3 + (1+487*y+3282*y^2+487*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 6*y + y^2)*x^2/2
+ (1 + 84*y + 84*y^2 + y^3)*x^3/3
+ (1 + 1820*y + 12870*y^2 + 1820*y^3 + y^4)*x^4/4
+ (1 + 53130*y + 3268760*y^2 + 3268760*y^3 + 53130*y^4 + y^5)*x^5/5 +...
in which the coefficients form A209330(n,k) = binomial(n^2, n*k).

Paul D. Hanna, Rows n = 0..30, flattened.

Cf. A209197, A167006, A206830, A209330 (log), A155200.

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

Column 1 equals A209197.
Row sums equal A167006.
Antidiagonal sums equal A206830.

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(""))

A209331 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, n*k-k^2).

Original entry on oeis.org

1, 1, 2, 7, 86, 1905, 66002, 5218373, 1340847046, 688750226335, 527838995308056, 707409447204872377, 2844096719471817175298, 30274246332924074325724393, 517646331335208169889265781259, 13363896516779950029547538703868509

Offset: 0

Paul D. Hanna, Mar 06 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..79

Cf. A209330, A167009, A238696, A209428.

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

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

Equals the antidiagonal sums of triangle A209330(n,k) = C(n^2,n*k).

Limit n->infinity a(n)^(1/n^2) = ((1-r)/r)^((1-r)^2/(3-4*r)) = 1.4360944969025357119535113523184471047971386419..., where r = A323777 = 0.220676041323740696312822269998050167187681031... is the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r). - Vaclav Kotesovec, Mar 03 2014

Name corrected by 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(""))

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

A245243 Triangle, read by rows, defined by T(n,k) = C(n^2 - k^2, n*k - k^2), for k=0..n, n>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 28, 10, 1, 1, 455, 495, 35, 1, 1, 10626, 54264, 8008, 126, 1, 1, 324632, 10518300, 4686825, 125970, 462, 1, 1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1, 1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1

Offset: 0

Paul D. Hanna, Jul 14 2014

Row sums equal A245242.

Central terms are A245245(n) = C(3*n^2, n^2).

			Triangle T(n,k) = C(n^2 - k^2, n*k - k^2) begins:
1;
1, 1;
1, 3, 1;
1, 28, 10, 1;
1, 455, 495, 35, 1;
1, 10626, 54264, 8008, 126, 1;
1, 324632, 10518300, 4686825, 125970, 462, 1;
1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1;
1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1; ...

Paul D. Hanna, Table of n, a(n) for rows 0..30 of flattened triangle.

Cf. A245242 (row sums), A245245 (central terms), A209330, A228832.

Table[Binomial[n^2-k^2,n k-k^2],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 06 2019 *)

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

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

T(n,k) = C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).
T(n,k) = (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).
T(n,k) = ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).

cp's OEIS Frontend

A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

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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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A206830 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * x^k ).

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A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ), as read by rows.

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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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A209331 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, n*k-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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A206830 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, nk) x^k ).

A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, nk) y^k ), as read by rows.

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