A167006 -id:A167006 - 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

A206850 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * x^k ).

Original entry on oeis.org

1, 1, 2, 4, 8, 56, 522, 5972, 424954, 16560881, 1528544877, 483389731955, 70609119680761, 53933819677734187, 58734216507052608587, 38789122414735365076327, 202547156817505166242299130, 712808848212730366850407506134, 2914935606380176735260119042755221

Offset: 0

Paul D. Hanna, Feb 13 2012

nonn

Equals antidiagonal sums of triangle A228902.

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...
such that, by definition, the logarithm equals the series:
log(A(x)) = x*(1+x) + x^2*(1 + 4*x + x^2)/2
+ x^3*(1 + 9*x + 126*x^2 + x^3)/3
+ x^4*(1 + 16*x + 1820*x^2 + 11440*x^3 + x^4)/4
+ x^5*(1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + x^5)/5
+ x^6*(1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + x^6)/6
+ x^7*(1 + 49*x + 211876*x^2 + 2054455634*x^3 + 3348108992991*x^4 + 63205303218876*x^5 + 262596783764*x^6 + x^7)/7 +...
+ x^n*(Sum_{k=0..n} binomial(n^2, k^2)*x^k)/n  +...

Cf. A206851 (log), A228902, A206830, A167006.

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

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.

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), ", "))

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),", "))

A207137 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k(n-k))x^k ).

Original entry on oeis.org

1, 1, 2, 4, 17, 171, 3171, 101741, 7181615, 1274607729, 428568152553, 223160743256395, 185627109707405932, 320952534083059792786, 1367454166673309618606950, 11078799748881429582280609036, 137939599816546528357634500253053, 2679390013936303204526656964298150849

Offset: 0

Paul D. Hanna, Feb 15 2012

nonn

The logarithmic derivative yields A207138.

Equals the antidiagonal sums of triangle A228900.

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 17*x^4 + 171*x^5 + 3171*x^6 +...
where the logarithm of the g.f. equals the l.g.f. of A207138:
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 51*x^4/4 + 761*x^5/5 + 17913*x^6/6 +...

Cf. A207138 (log), A207135, A228900, A206850, A206830, A167006.

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

A228809 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) ).

Original entry on oeis.org

1, 2, 4, 12, 94, 2195, 158904, 31681195, 13904396167, 15305894726347, 44888344014554903, 288228807835914177564, 4270880356112396772814732, 169380654509201278629725097906, 15394658527137259981745081997280638, 3042352591056504014301304188228238554499

Offset: 0

Paul D. Hanna, Sep 04 2013

nonn

Logarithmic derivative equals A228808.

Equals row sums of triangle A228904.

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 +...
where
log(A(x)) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 + 927100*x^6/6 +...+ A228808(n)*x^n/n +...

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

Cf. A228808, A207135, A228904.

Cf. variants: A167006, A206848.

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

A167007 G.f.: A(x) = exp( Sum_{n>=1} A167010(n)*x^n/n ) where A167010(n) = Sum_{k=0..n} binomial(n,k)^n.

Original entry on oeis.org

1, 2, 5, 26, 501, 42262, 14564184, 18926665052, 96371663657380, 1825266130738144920, 136764680697906838980633, 38133043109557952095731186822, 42464330390232136488003531922964743

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 501*x^4 + 42262*x^5 + ...
log(A(x)) = 2*x + 6*x^2/2 + 56*x^3/3 + 1810*x^4/4 + 206252*x^5/5 + 86874564*x^6/6 + ... + A167010(n)*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..59

Cf. A155200, A167006, A167010.

A167010:= func< n | (&+[Binomial(n,j)^n: j in [0..n]]) >;
function A167007(n)
  if n lt 2 then return n+1;
  else return (&+[A167010(j)*A167007(n-j): j in [1..n]])/n;
  end if; return A167007;
end function;
[A167007(n): n in [0..20]]; // G. C. Greubel, Aug 26 2022

A167010[n_]:= A167010[n]= Sum[Binomial[n,j]^n, {j,0,n}];
A167007[n_]:= A167007[n]= If[n==0, 1, (1/n)*Sum[A167010[j]*A167007[n-j], {j,n}]];
Table[A167007[n], {n,0,30}] (* G. C. Greubel, Aug 26 2022 *)

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

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sum(j=0, k, binomial(k, j)^k)*a(n-k)))} \\ Paul D. Hanna, Nov 25 2009

def A167010(n): return sum(binomial(n,j)^n for j in (0..n))
def A167007(n): return 1 if (n==0) else (1/n)*sum( A167010(j)*A167007(n-j) for j in (1..n))
[A167007(n) for n in (0..30)] # G. C. Greubel, Aug 26 2022

a(n) = (1/n)*Sum_{k=1..n} A167010(k)*a(n-k) for n>0 with a(0)=1. - Paul D. Hanna, Nov 25 2009

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

cp's OEIS Frontend

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

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A206850 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * x^k ).

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

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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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A207137 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))*x^k ).

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A228809 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} 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.

A207137 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k(n-k))x^k ).

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.