A166894 -id:A166894 - cp's OEIS frontend

A166895 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k), n>=1.

1, 3, 7, 39, 366, 5697, 194881, 16288695, 2430565261, 564615230758, 257227244037248, 319346787227133873, 832952161388710135215, 3382434539389226013260403, 22966972221673234523620345857

Offset: 1

Paul D. Hanna, Nov 23 2009

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 366*x^5/5 + 5697*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 89*x^5 + 1050*x^6 +...+ A166894(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 1..50

Cf. A166894, A209428.

Table[Sum[Binomial[n - k, k]^(n - k) *n/(n - k), {k, 0, Floor[n/2]}], {n, 1, 25}] (* G. C. Greubel, May 27 2016 *)

a(n)=sum(k=0,n\2,binomial(n-k,k)^(n-k)*n/(n-k))

Logarithmic derivative of A166894.

Limit_{n->oo} a(n)^(1/n^2) = (1/r - 1)^((1 - r)^2/(3 - 4*r)) = 1.436094496902535711953511352318447104797138641971237143543..., where r = A323777 = 0.220676041323740696312822269998050167187681031027574... is the root of the equation (1 - 2*r)^(3 - 4*r) = (1 - r)^(2*(1 - r))*r^(1 - 2*r). - Vaclav Kotesovec, Nov 20 2024

A181080 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k+1) * x^k] * x^n/n ).

Original entry on oeis.org

1, 1, 2, 4, 14, 83, 774, 10641, 255918, 14643874, 1752083557, 320079087261, 79294841767020, 27407454296637142, 16895839815165609994, 26064121763003372842186, 82824096391548076720149081

Offset: 0

Paul D. Hanna, Oct 02 2010

nonn

Conjecture: this sequence consists entirely of integers.
Note that the following g.f. does NOT yield an integer series:
  exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k) * x^k] * x^n/n ).

			G.f. A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 +...
The logarithm of g.f. A(x) begins:
  log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + 4077*x^6/6 + ... + A181081(n)*x^n/n + ...
and equals the series:
  log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
  + (1 + 3^3*x +  3^2*x^2 +      x^3)*x^3/3
  + (1 + 4^4*x +  6^3*x^2 +  4^2*x^3 +      x^4)*x^4/4
  + (1 + 5^5*x + 10^4*x^2 + 10^3*x^3 +  5^2*x^4 +     x^5)*x^5/5
  + (1 + 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..90

Variants: A166894, A181070, A181082.

Cf. A181081 (log).

m:=20;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( Exp( (&+[ (&+[ Binomial(n,k)^(n-k+1)*x^(n+k)/n : k in [0..n]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021

With[{m=20}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n, k]^(n-k+1)*x^(n+k)/n, {k,0,n}], {n,m+1}]], {x,0,m}], x]] (* G. C. Greubel, Apr 05 2021 *)

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

m=20;
def A181066_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( exp( sum( sum( binomial(n,k)^(n-k+1)*x^(n+k)/n for k in (0..n) ) for n in (1..m+1)) ) ).list()
A181066_list(m) # G. C. Greubel, Apr 05 2021

A166896 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k] * x^n/n ), an integer series in x.

Original entry on oeis.org

1, 1, 2, 6, 16, 45, 142, 459, 1508, 5122, 17787, 62649, 223971, 811339, 2970032, 10974150, 40893393, 153512844, 580082454, 2205046961, 8427087958, 32362949488, 124837337235, 483508287359, 1879669861074, 7332469937755

Offset: 0

Paul D. Hanna, Nov 23 2009

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 + 1765*x^7/7 +...+ A166897(n)*x^n/n +...

Cf. A166897, variants: A166894, A166898.

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3*x^k)*x^m/m)+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^3*m/(m-k))*x^m/m)+x*O(x^n)), n)}

G.f.: exp( Sum_{n>=1} A166897(n)*x^n/n ) where A166897(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k).

A166898 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4 * x^k] * x^n/n ), an integer series in x.

Original entry on oeis.org

1, 1, 2, 10, 38, 137, 646, 3241, 15623, 79439, 427562, 2317396, 12715372, 71543343, 408543758, 2353591560, 13717994046, 80827739181, 480016288156, 2871701561720, 17304832805996, 104933348346951, 639814473417775

Offset: 0

Paul D. Hanna, Nov 23 2009

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 38*x^4 + 137*x^5 + 646*x^6 + 3241*x^7 +...
log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 111*x^4/4 + 456*x^5/5 + 2697*x^6/6 + 15961*x^7/7 +...+ A166899(n)*x^n/n +...

Cf. A166897, variants: A166894, A166898.

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^4*x^k)*x^m/m)+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^4*m/(m-k))*x^m/m)+x*O(x^n)), n)}

G.f.: exp( Sum_{n>=1} A166899(n)*x^n/n ) where A166899(n) = Sum_{k=0..[n/2]} C(n-k,k)^4*n/(n-k).

A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 76, 347, 76, 1, 1, 701, 20429, 20429, 701, 1, 1, 8477, 1919660, 10707908, 1919660, 8477, 1, 1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1, 1, 2223278, 47484618291, 12099129236936, 72078431500368

Offset: 0

Paul D. Hanna, Mar 08 2012

Column 1 is A060946.
Column 2 is A209425.
Row sums equal A167007.
Antidiagonal sums equal A166894.
Central terms form A209426.

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 12, 12, 1;
1, 76, 347, 76, 1;
1, 701, 20429, 20429, 701, 1;
1, 8477, 1919660, 10707908, 1919660, 8477, 1;
1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1;
1, 2223278, 47484618291, 12099129236936, 72078431500368, 12099129236936, 47484618291, 2223278, 1; ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+12*y+12*y^2+y^3)*x^3 + (1+76*y+20429*y^2+76*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^2*y + y^2)*x^2/2
+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
+ (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
in which the coefficients are found in triangle A209427.

Paul D. Hanna, Table of n, a(n) for n = 0..495 for Rows 0..30 of this triangle in flattened form.

Cf. A060946, A209425, A167007, A166894, A209426, A209427, A209196.

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

cp's OEIS Frontend

A166895 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k), n>=1.

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A181080 Expansion of g.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^(n-k+1) * x^k] * x^n/n ).

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A166896 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k] * x^n/n ), an integer series in x.

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A166898 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4 * x^k] * x^n/n ), an integer series in x.

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A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.

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