A217668 -id:A217668 - cp's OEIS frontend

A326004 G.f.: Sum_{n>=0} (n+1)(n+2)(n+3)/3! * x^n * (1 + x^n)^n.

1, 4, 14, 20, 55, 56, 154, 120, 305, 280, 566, 364, 1189, 560, 1520, 1376, 2429, 1140, 4570, 1540, 5226, 4544, 6304, 2600, 14685, 3556, 10934, 11980, 18215, 4960, 31882, 5984, 31289, 27160, 27150, 12636, 82093, 9880, 39920, 55160, 93631, 13244, 121178, 15180, 126875, 130696, 78224, 19600, 316645, 22940, 165386, 179844, 281399, 27720, 370090, 150976, 410629, 297560, 179830, 37820, 1208458, 41664, 229184, 489280, 801305, 450516, 987482, 54740

Offset: 0

Paul D. Hanna, Jun 01 2019

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 4 and p = 1, q = x, r = x.

			G.f.: A(x) = 1 + 4*x + 14*x^2 + 20*x^3 + 55*x^4 + 56*x^5 + 154*x^6 + 120*x^7 + 305*x^8 + 280*x^9 + 566*x^10 + 364*x^11 + 1189*x^12 + 560*x^13 + 1520*x^14 + 1376*x^15 + 2429*x^16 + 1140*x^17 + 4570*x^18 + 1540*x^19 + 5226*x^20 + ...
where we have the following series identity:
A(x) = 1 + 4*x*(1+x) + 10*x^2*(1+x^2)^2 + 20*x^3*(1+x^3)^3 + 35*x^4*(1+x^4)^4 + 56*x^5*(1+x^5)^5  + 84*x^6*(1+x^6)^6 + 120*x^7*(1+x^7)^7 + 165*x^8*(1+x^8)^8 + 220*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^4 + 4*x^2/(1-x^2)^5 + 10*x^6/(1-x^3)^6 + 20*x^12/(1-x^4)^7 + 35*x^20/(1-x^5)^8 + 56*x^30/(1-x^6)^9 + 84*x^42/(1-x^7)^10 + 120*x^56/(1-x^8)^11 +...

Cf. A217668 (k=1), A326002 (k=2), A326003 (k=3), A326005 (k=5).

{a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)/3! * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)/3! * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+4))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Generating functions.
(1) Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^(n*(n+1)) / (1 - x^(n+1))^(n+4).

A326005 G.f.: Sum_{n>=0} (n+1)(n+2)(n+3)(n+4)/4! x^n * (1 + x^n)^n.

Original entry on oeis.org

1, 5, 20, 35, 100, 126, 330, 330, 775, 820, 1631, 1365, 3535, 2380, 5370, 5136, 9085, 5985, 16900, 8855, 21966, 19580, 29965, 17550, 60375, 24381, 58345, 57205, 90350, 40920, 152837, 52360, 164145, 141120, 175560, 93801, 404500, 101270, 280175, 309050, 503041, 148995, 714435, 178365, 748705, 708946, 633950, 249900, 1771645, 295135, 1120236, 1155015, 1760500, 395010, 2483110, 905576, 2622545, 2036060, 1744525, 595665, 6962328, 677040, 2343880

Offset: 0

Paul D. Hanna, Jun 01 2019

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 5 and p = 1, q = x, r = x.

			G.f.: A(x) = 1 + 5*x + 20*x^2 + 35*x^3 + 100*x^4 + 126*x^5 + 330*x^6 + 330*x^7 + 775*x^8 + 820*x^9 + 1631*x^10 + 1365*x^11 + 3535*x^12 + 2380*x^13 + 5370*x^14 + 5136*x^15 + 9085*x^16 + 5985*x^17 + 16900*x^18 + 8855*x^19 + 21966*x^20 + ...
where we have the following series identity:
A(x) = 1 + 5*x*(1+x) + 15*x^2*(1+x^2)^2 + 35*x^3*(1+x^3)^3 + 70*x^4*(1+x^4)^4 + 126*x^5*(1+x^5)^5  + 210*x^6*(1+x^6)^6 + 330*x^7*(1+x^7)^7 + 495*x^8*(1+x^8)^8 + 715*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^5 + 5*x^2/(1-x^2)^6 + 15*x^6/(1-x^3)^7 + 35*x^12/(1-x^4)^8 + 70*x^20/(1-x^5)^9 + 126*x^30/(1-x^6)^10 + 210*x^42/(1-x^7)^11 + 330*x^56/(1-x^8)^12 +...

Cf. A217668 (k=1), A326002 (k=2), A326003 (k=3), A326004 (k=4).

{a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)*(m+4)/4! * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, (m+1)*(m+2)*(m+3)*(m+4)/4! * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+5))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Generating functions.
(1) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^(n*(n+1)) / (1 - x^(n+1))^(n+5).

A327249 Expansion of Sum_{k>=1} x^k * (1 + k * x^k)^k.

Original entry on oeis.org

1, 2, 1, 5, 1, 14, 1, 17, 28, 26, 1, 160, 1, 50, 251, 321, 1, 622, 1, 1607, 1030, 122, 1, 6257, 3126, 170, 2917, 12202, 1, 27291, 1, 28929, 6656, 290, 84036, 117721, 1, 362, 13183, 407121, 1, 417881, 1, 220100, 850312, 530, 1, 2246465, 823544, 2100626

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

nonn

Cf. A006005 (positions of 1's), A087909, A217668, A260180, A327238.

[&+[(n div d)^(d-1)*Binomial(n div d,d-1):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Sep 15 2019

nmax = 50; CoefficientList[Series[Sum[x^k (1 + k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^(# - 1) Binomial[n/#, # - 1] &], {n, 1, 50}]

a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d,d-1)); \\ Michel Marcus, Sep 15 2019

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d,d-1).

A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.

Original entry on oeis.org

2, 4, 2, 10, 2, 12, 16, 16, 2, 50, 24, 24, 58, 28, 72, 154, 2, 36, 302, 40, 142, 270, 332, 48, 242, 472, 574, 614, 60, 60, 2282, 64, 2, 1454, 1362, 2494, 628, 76, 1940, 3304, 3642, 84, 5266, 88, 662, 13180, 3544, 96, 994, 6106, 14292, 13602, 3434, 108, 8102, 14854, 16018, 24778, 7310, 120, 35684

Offset: 0

Paul D. Hanna, Jul 23 2015

nonn

Compare to the curious identities:
(1) Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.
(2) Sum_{n=-oo..+oo} (-x)^n * (1 + x^n)^n  =  0.

			G.f.: A(x) = 2 + 4*x^2 + 2*x^4 + 10*x^6 + 2*x^8 + 12*x^10 + 16*x^12 + 16*x^14 + 2*x^16 + 50*x^18 + 24*x^20 +...
where A(x) = 1 + P(x) + N(x) with
P(x) = x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5 +...
N(x) = 1/(1+x) + x^2/(1+x^2)^2 + x^6/(1+x^3)^3 + x^12/(1+x^4)^4 + x^20/(1+x^5)^5 +...
Explicitly,
P(x) = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 + 4*x^9 + 6*x^10 + x^11 + 14*x^12 + x^13 + 8*x^14 + 11*x^15 + 13*x^16 + x^17 + 25*x^18 + x^19 + 22*x^20 + 22*x^21 + 12*x^22 + x^23 + 61*x^24 + 6*x^25 +...+ A217668(n)*x^n +...
N(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 - x^11 + 2*x^12 - x^13 + 8*x^14 - 11*x^15 - 11*x^16 - x^17 + 25*x^18 - x^19 + 2*x^20 - 22*x^21 + 12*x^22 - x^23 - 3*x^24 - 6*x^25 +...+ A260148(n)*x^n +...

Cf. A260147, A217668, A260148, A363569, A363558, A363559, A363561.

terms = 100; max = 2 terms; Sum[x^n*(1 + x^n)^n, {n, -max, max}] + O[x]^max // CoefficientList[#, x^2]& (* Jean-François Alcover, May 16 2017 *)

{a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^k*(1+x^k)^k + O(x^(2*n+2)) ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))

{a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^(k^2-k) / (1 + x^k)^k  + O(x^(2*n+2)) ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))

{a(n) = local(A=1); A = sum(k=1, sqrtint(2*n)+2, x^(k^2-k) *((1 + x^k)^k + (1 - x^k)^k) / (1 - x^(2*k)  + O(x^(2*n+2)) )^k ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))

{a(n) = local(A=1); A = sum(k=-sqrtint(n)-1, n+1, x^k*((1+x^k)^(2*k) + (1-x^k)^(2*k)) + O(x^(n+1)) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

G.f.: Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n.
G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n.
G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n.
G.f.: Sum_{n>=1} x^(n^2-n) *((1 + x^n)^n + (1 - x^n)^n) / (1 - x^(2*n))^n.
G.f.: Sum_{n=-oo..+oo} x^n * ((1 + x^n)^(2*n) + (1 - x^n)^(2*n))  =  Sum_{n>=0} a(n)*x^n.
a(n) = 2*A260147(n).
a(2^n) = 2 for n > 0 (conjecture).
a(p) = 2*p+2 for primes p > 3 (conjecture).

A266330 Triangle, read by rows, of the coefficients in the g.f.: Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n.

Original entry on oeis.org

-1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 0, 0, 0, 1, -1, 2, 0, -2, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 3, -1, 1, -3, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 4, 0, 0, 0, -4, 0, 0, 0, 1, -1, 0, -3, 0, 3, 0, 0, 0, 0, 0, 1, -1, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 6, -6, 1, -1, 6, 0, -6, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 1, -1, 0, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 8, 0, 4, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 9, -15, 0, 0, 0, 0, 15, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul D. Hanna, Dec 27 2015

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
Note that the g.f.:
A(x,y) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n
may be written
A(x,y) = Sum_{n>=0} R(n,y) * x^n / y^(n+1)
such that row polynomials R(n,y) consist of square powers of y:
R(n,y) = Sum_{k=0..n+1} T(n,k) * y^(k^2).

			This triangle of coefficients T(n,k) begins:
  -1, 1;
  -1, 0, 1;
  -1, 1, -1, 1;
  -1, 0, 0, 0, 1;
  -1, 2, 0, -2, 0, 1;
  -1, 0, 0, 0, 0, 0, 1;
  -1, 3, -1, 1, -3, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 4, 0, 0, 0, -4, 0, 0, 0, 1;
  -1, 0, -3, 0, 3, 0, 0, 0, 0, 0, 1;
  -1, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 6, -6, 1, -1, 6, 0, -6, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 8, 0, 4, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 9, -15, 0, 0, 0, 0, 15, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 1; ...
in which the g.f. of column k > 0 is given by:
z^(k-1)*(1 - z^(k-1))^(k-1) + z^(k*(k+1))/(z^(k+1) - 1)^(k+1).
...
G.f.: A(x,y) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n may be written as
A(x,y) = Sum_{n>=0} R(n,y) * x^n / y^(n+1), where row polynomials R(n,y) consist of square powers of y:
R(n,y) = Sum_{k=0..n+1} T(n,k) * y^(k^2);
this triangle lists the coefficients of y^(k^2) in R(n,y), which begin:
  R(0,y) = y - 1;
  R(1,y) = y^4 - 1;
  R(2,y) = y^9 - y^4 + y - 1;
  R(3,y) = y^16 - 1;
  R(4,y) = y^25 - 2*y^9 + 2*y - 1;
  R(5,y) = y^36 - 1;
  R(6,y) = y^49 - 3*y^16 + y^9 - y^4 + 3*y - 1;
  R(7,y) = y^64 - 1;
  R(8,y) = y^81 - 4*y^25 + 4*y - 1;
  R(9,y) = y^100 + 3*y^16 - 3*y^4 - 1;
  R(10,y) = y^121 - 5*y^36 + 5*y - 1;
  R(11,y) = y^144 - 1;
  R(12,y) = y^169 - 6*y^49 + 6*y^25 - y^16 + y^9 - 6*y^4 + 6*y - 1;
  R(13,y) = y^196 - 1;
  R(14,y) = y^225 - 7*y^64 + 7*y - 1;
  R(15,y) = y^256 + 10*y^36 - 10*y^4 - 1;
  R(16,y) = y^289 - 8*y^81 - 4*y^25 + 4*y^9 + 8*y - 1;
  R(17,y) = y^324 - 1;
  R(18,y) = y^361 - 9*y^100 + 15*y^49 - 15*y^4 + 9*y - 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..5251, for rows 0..100 in flattened form of this triangle.

Cf. A217668, A260147 (y=-1).

/* Prints rows 0..50 of this triangle: */
{SUM=sum(n=-51,51,x^n*y^n*(y^n-x^n +O(x^51))^n); V=Vec(SUM);
T(n,k)=polcoeff(V[n+1]*y^(n+1) + y*O(y^((n+1)^2)), k^2)}
for(n=0,50,for(k=0,n+1,print1( T(n,k), ", "));print(""))

/* Quick print of row polynomials (informal): */
{SUM=sum(n=-51,51,x^n*y^n*(y^n-x^n +O(x^51))^n); V=Vec(SUM);
for(n=1,50,print("R("n-1",y) = "V[n]*y^n";")) }

/* Compare these sums (informal sanity check): */
Axy = sum(n=-16,16, x^n*y^n*(y^n-x^n +O(x^16))^n )
Axy = -(1/y)/(1-x/y) + sum(n=1,15, y^(n^2-1) * ( (x/y)^(n-1)*(1 - (x/y)^(n-1))^(n-1) + (x/y)^(n*(n+1)) / ((x/y)^(n+1) - 1)^(n+1) ) +O(x^16) )

G.f.: -(1/y)/(1-z) + (1/y) * Sum_{n>=1} y^(n^2) * ( z^(n-1)*(1 - z^(n-1))^(n-1) + z^(n*(n+1)) / (z^(n+1) - 1)^(n+1) ) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n, where z = x/y.
Row sums are all zeros.
Row sums of absolute values of terms yield 2 * A217668(n) for row n>=0.
Sum_{k=0..2*n+1} (-1)^k * T(2*n,k) = (-2) * A260147(n) for n>=0.
Sum_{k=0..2*n+2} (-1)^k * T(2*n+1,k) = 0 for n>=0.
Sum_{k=0..2*n+1} I^(k^2) * T(2*n,k) = (I-1) * A260147(n) for n>=0, where I^2 = -1.
Sum_{k=0..2*n+2} I^(k^2) * T(2*n+1,k) = 0 for n>=0, where I^2 = -1.

A360771 Expansion of Sum_{k>=0} (x * (2 + x^k))^k.

Original entry on oeis.org

1, 2, 5, 8, 20, 32, 77, 128, 288, 518, 1104, 2048, 4313, 8192, 16832, 32848, 66568, 131072, 264688, 524288, 1053737, 2097824, 4205568, 8388608, 16803744, 33554442, 67162112, 134222336, 268550704, 536870912, 1073999165, 2147483648, 4295493376, 8589962752

Offset: 0

Seiichi Manyama, Feb 20 2023

nonn

Cf. A217668, A260148.

a[n_] := DivisorSum[n, 2^(# - n/# + 1) * Binomial[#, n/# - 1] &]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Aug 02 2023 *)

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(2+x^k))^k))

a(n) = if(n==0, 1, sumdiv(n, d, 2^(d-n/d+1)*binomial(d, n/d-1)));

a(n) = Sum_{d|n} 2^(d-n/d+1) * binomial(d,n/d-1) for n > 0.

If p is an odd prime, a(p) = 2^p.

A370668 Expansion of Sum_{k>0} k! * ( x * (1+x^k) )^k.

Original entry on oeis.org

1, 3, 6, 28, 120, 740, 5040, 40416, 362898, 3629400, 39916800, 479006070, 6227020800, 87178326480, 1307674369200, 20922790210656, 355687428096000, 6402373709004720, 121645100408832000, 2432902008212929224, 51090942171709545840, 1124000727778046764800

Offset: 1

Seiichi Manyama, Feb 25 2024

nonn

Cf. A240172, A370509, A370510.

Cf. A217668.

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*(x*(1+x^k))^k))

a(n) = sumdiv(n,d, d!*binomial(d, n/d-1));

a(n) = Sum_{d|n} d! * binomial(d,n/d-1).

If p is an odd prime, a(p) = p!.

cp's OEIS Frontend

A326004 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * (1 + x^n)^n.

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A326005 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n.

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A327249 Expansion of Sum_{k>=1} x^k * (1 + k * x^k)^k.

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A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.

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A266330 Triangle, read by rows, of the coefficients in the g.f.: Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n.

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A360771 Expansion of Sum_{k>=0} (x * (2 + x^k))^k.

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A370668 Expansion of Sum_{k>0} k! * ( x * (1+x^k) )^k.

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PARI

Formula

A326004 G.f.: Sum_{n>=0} (n+1)(n+2)(n+3)/3! * x^n * (1 + x^n)^n.

A326005 G.f.: Sum_{n>=0} (n+1)(n+2)(n+3)(n+4)/4! x^n * (1 + x^n)^n.