cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A291937 G.f.: Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n.

Original entry on oeis.org

1, 2, 0, 4, -3, 6, -3, 8, -15, 28, -24, 12, 0, 14, -48, 96, -95, 18, 55, 20, -180, 232, -120, 24, -35, 76, -168, 460, -580, 30, 515, 32, -927, 804, -288, 456, -497, 38, -360, 1288, -1169, 42, 847, 44, -2958, 3700, -528, 48, -2599, 148, 2526, 2772, -5537, 54, 595, 5336, -6930, 3820, -840, 60, -791, 62, -960, 6448, -12351, 12936, -3167, 68, -15435, 6648, 21365, 72, -26646, 74, -1368, 35776, -23730, 8394, -16548, 80, 7101
Offset: 0

Views

Author

Paul D. Hanna, Sep 06 2017

Keywords

Comments

Compare o.g.f. to: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
Compare l.g.f. to: Sum_{n=-oo..+oo, n<>0} x^n * (1 - x^(n-1))^n / n = -log(1-x).
Whenever a(n+2) is a multiple of n > 7, then a(n+2)/n = -(n+4)/4, with very few exceptions (n = 18, 131, 412, ... and n = 10, a(12) = 0). In particular, when n-1 is a prime of the form p = 4k + 3, then a(p+3) = -(k+2)(p+1) (as compared to a(p) = p+1), except for k = 11, 16, 26, 31, 37, 41, .... What exactly are these exceptions? - M. F. Hasler, Oct 10 2017

Examples

			O.g.f.: A(x) = 1 + 2*x + 4*x^3 - 3*x^4 + 6*x^5 - 3*x^6 + 8*x^7 - 15*x^8 + 28*x^9 - 24*x^10 + 12*x^11 + 14*x^13 - 48*x^14 + 96*x^15 - 95*x^16 + 18*x^17 + 55*x^18 + 20*x^19 - 180*x^20 + 232*x^21 - 120*x^22 + 24*x^23 - 35*x^24 + 76*x^25 - 168*x^26 + 460*x^27 - 580*x^28 + 30*x^29 + 515*x^30 +...
where A(x) = P(x) + Q(x) with
P(x) = x*(1-x) + 2*x^2*(1-x^2)^2 + 3*x^3*(1-x^3)^3 + 4*x^4*(1-x^4)^4 + 5*x^5*(1-x^5)^5 +...+ n * x^n * (1 + x^n)^n + ...
Q(x) = 1/(1-x) - 2*x^2/(1-x^2)^2 + 3*x^6/(1-x^3)^3 - 4*x^12/(1-x^4)^4 + 5*x^20/(1-x^5)^5 + ... + -(-1)^n * n * x^(n^2-n) / (1 - x^n)^n + ...
Explicitly,
P(x) = x + x^2 + 3*x^3 + 5*x^5 - x^6 + 7*x^7 - 8*x^8 + 18*x^9 - 15*x^10 + 11*x^11 - 3*x^12 + 13*x^13 - 35*x^14 + 65*x^15 - 64*x^16 + 17*x^17 + 27*x^18 + 19*x^19 - 126*x^20 + 168*x^21 - 99*x^22 + 23*x^23 - 16*x^24 + 50*x^25 - 143*x^26 + 351*x^27 - 413*x^28 + 29*x^29 + 340*x^30 + ...
Q(x) = 1 + x - x^2 + x^3 - 3*x^4 + x^5 - 2*x^6 + x^7 - 7*x^8 + 10*x^9 - 9*x^10 + x^11 + 3*x^12 + x^13 - 13*x^14 + 31*x^15 - 31*x^16 + x^17 + 28*x^18 + x^19 - 54*x^20 + 64*x^21 - 21*x^22 + x^23 - 19*x^24 + 26*x^25 - 25*x^26 + 109*x^27 - 167*x^28 + x^29 + 175*x^30 + ...
Also, A(x) = M(x) + N(x) with
M(x) = x^2 + 4*x^4*(1-x^2) + 9*x^6*(1-x^3)^2 + 16*x^8*(1-x^4)^3 + 25*x^10*(1-x^5)^4 + ... + n^2 * x^(2*n) * (1 - x^n)^(n-1) + ...
N(x) = 1/(1-x)^2 - 4*x^2/(1-x^2)^3 + 9*x^6/(1-x^3)^4 - 16*x^12/(1-x^4)^5 + 25*x^20/(1-x^5)^6 + ... + -(-1)^n * n^2 * x^(n^2-n) / (1 - x^n)^(n+1) + ...
Explicitly,
M(x) = x^2 + 4*x^4 + 5*x^6 + 16*x^8 - 18*x^9 + 25*x^10 - 3*x^12 + 49*x^14 - 100*x^15 + 112*x^16 - 99*x^18 + 234*x^20 - 294*x^21 + 121*x^22 + 56*x^24 - 100*x^25 + 169*x^26 - 648*x^27 + 931*x^28 - 1010*x^30 + ...
N(x) = 1 + 2*x - x^2 + 4*x^3 - 7*x^4 + 6*x^5 - 8*x^6 + 8*x^7 - 31*x^8 + 46*x^9 - 49*x^10 + 12*x^11 + 3*x^12 + 14*x^13 - 97*x^14 + 196*x^15 - 207*x^16 + 18*x^17 + 154*x^18 + 20*x^19 - 414*x^20 + 526*x^21 - 241*x^22 + 24*x^23 - 91*x^24 + 176*x^25 - 337*x^26 + 1108*x^27 - 1511*x^28 + 30*x^29 + 1525*x^30 + ...
Terms at powers of 2 begin:
a(2^n) = [2, 0, -3, -15, -95, -927, -12351, -457215, -137484287, -71927383551, -12774376215944191, -2073810501234874519551, -78004011261694477161745918353407, ...].
Terms at powers of 3 begin:
a(3^n) = [2, 4, 28, 460, 10774, 80195104, 2894790054826, ..., A292184(n), ...].
Terms at powers of 5 begin:
a(5^n) = [2, 6, 76, 379626, 1259880626, 4828768869002981409762696876, ...].

Links

Crossrefs

Cf. A292184, A262007, A292177, A293129.

Programs

  • Mathematica
    terms = 200; Sum[n*x^n*(1 - x^n)^n, {n, -terms, terms}] + O[x]^terms //
CoefficientList[#, x]& (* Jean-François Alcover, Oct 11 2017 *)
  • PARI
    {a(n)=my(l=1+O(x^(2*n+2))); polcoeff(sum(k=-n-2, n+2, k*x^k*(l-x^k)^k), n)} \\ Edited by M. F. Hasler, Oct 11 2017
  • PARI
    {a(n) = my(l=1+O(x^(2*n+2))); polcoeff(sum(k=-n-2, n+2, if(k, k^2 * x^(2*k) * (l - x^k)^(k-1))), n)} \\ Edited by M. F. Hasler, Oct 11 2017
  • PARI
    {a(n) = my(x='x+O('x^(2*n+2))); polcoeff(sum(k=-n-2, sqrtint(2*n)+2, -(-1)^k * k * x^(k^2-k) / (1 - x^k)^k), n)} \\ Edited by M. F. Hasler, Oct 11 2017
  • PARI
    {a(n) = my(x='x+O('x^(2*n+2))); polcoeff( sum(k=-n-2, sqrtint(2*n), if(k, -(-1)^k * k * x^(k^2-k) / (1 - x^k)^(k+1) )), n)} \\ Edited by M. F. Hasler, Oct 11 2017
for(n=0, 80, print1(a(n), ", "))
  • PARI
    A291937_vec(n)={my(x='x+O('x^(2*n+2))); -Vec(sum(k=-n-2, sqrtint(2*n), if(k,(-1)^k*k*x^(k^2-k)/(1-x^k)^(k+1))))[1..n+1]} \\ In case several values in a(0..n) are required, it is most efficient to compute the whole vector at once. E.g., sum(n=0..150,a(n)) takes ~ 10 sec., vecsum(A291937_vec(150)) takes ~ 0.1 sec. - M. F. Hasler, Oct 11 2017

Formula

The o.g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n.
(2) A(x) = Sum_{n=-oo..+oo} n^2 * x^(2*n) * (1 - x^n)^(n-1).
(3) A(x) = Sum_{n=-oo..+oo} -n * x^(2*n) * (1 - x^n)^(n-1).
(4) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n * x^(n^2-n) / (1 - x^n)^n.
(5) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n^2 * x^(n^2-n) / (1 - x^n)^(n+1).
(6) A(x) = Limit_{k->oo} Sum_{n=-oo..+oo} x^(n-k) * (1 - x^n - x^(n+k))^n.
(7) A(x) = Limit_{k->oo} Sum_{n=-oo..+oo} x^(n-k) * (1 - x^n + n*x^(n+k))^n.
The l.g.f. L(x) = Sum_{n>=1} a(n) * x^n / n satisfies:
(8) L(x) = -1 + Sum_{n=-oo..+oo, n<>0} x^n * (1 - x^n)^n / n.
a(p) = p+1 for odd primes p.

A293597 L.g.f.: Sum_{n>=1} (x - x^(2*n-1))^(2*n-1) / (2*n-1).

Original entry on oeis.org

0, 1, -4, 8, -11, 1, 14, 1, -50, 58, 1, 1, -54, 1, -28, 311, -340, 1, 75, 1, -81, 345, -44, 1, -1427, 1531, -52, 496, -1253, 1, 1343, 1, -2924, 738, -68, 9444, -10073, 1, -76, 1028, 3691, 1, -4691, 1, -6941, 21295, -92, 1, -55580, 33034, 28180, 1752, -11479, 1, -54063, 42847, 19437, 2186, -116, 1, -77934, 1, -124, 238507, -169032, 85151, -188859, 1, -25755, 3198, 432636, 1, -513328, 1, -148, 157041, -36005, 711327, -465347, 1
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2017

Keywords

Examples

			L.g.f.: A(x) = x^3/3 - 4*x^5/5 + 8*x^7/7 - 11*x^9/9 + x^11/11 + 14*x^13/13 + x^15/15 - 50*x^17/17 + 58*x^19/19 + x^21/21 + x^23/23 - 54*x^25/25 + x^27/27 - 28*x^29/29 + 311*x^31/31 - 340*x^33/33 + x^35/35 + 75*x^37/37 + x^39/39 - 81*x^41/41 + 345*x^43/43 - 44*x^45/45 + x^47/47 - 1427*x^49/49 + 1531*x^51/51 - 52*x^53/53 + 496*x^55/55 - 1253*x^57/57 + x^59/59 + 1343*x^61/61 + x^63/63 - 2924*x^65/65 +...
which may be written as
A(x) = (x - x) + (x - x^3)^3/3 + (x - x^5)^5/5 + (x - x^7)^7/7 + (x - x^9)^9/9 + (x - x^11)^11/11 + (x - x^13)^13/13 + (x - x^15)^15/15 + (x - x^17)^17/17 + (x - x^19)^19/19 + (x - x^21)^21/21 +...+ (x - x^(2*n-1))^(2*n-1)/(2*n-1) +...
The coefficient of x^(2^n+1)/(2^n+1) in A(x) for n>=1 begins:
[1, -4, -11, -50, -340, -2924, -169032, -33445208, -21619038032, 1 - A293599(n), ...].

Links

Crossrefs

Cf. A293129, A293598, A293599.

Programs

  • PARI
    {a(n) = my(A, Ox = O(x^(2*n+1)));
A = sum(m=1, n+1, (x - x^(2*m-1) +Ox)^(2*m-1) / (2*m-1) );
(2*n-1)*polcoeff(A, 2*n-1)}
for(n=1, 80, print1(a(n), ", "))

A293598 L.g.f.: Sum_{n>=1} x^((2*n-1)^2) / ( (2*n-1) * (1 - x^(2*n))^(2*n-1) ).

Original entry on oeis.org

1, 3, 5, 7, 12, 11, 26, 15, 51, 19, 91, 23, 155, 27, 232, 62, 341, 35, 592, 39, 656, 344, 870, 47, 1820, 51, 1431, 1441, 1843, 59, 4758, 63, 2925, 4489, 3197, 71, 11899, 75, 4466, 11376, 7650, 83, 23052, 87, 12816, 25025, 7936, 95, 57133, 99, 10706, 49131, 37220, 107, 79570, 2146, 62828, 89263, 15951, 119, 228096, 123, 19500, 152146, 169033, 18864, 218253, 135, 267972, 246308, 75153, 143, 724159, 147, 33227, 490146, 629034, 155, 512448, 159
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2017

Keywords

Examples

			L.g.f.: A(x) = x + 3*x^3/3 + 5*x^5/5 + 7*x^7/7 + 12*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + 51*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + 341*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + 2925*x^65/65 +...
which may be written as
A(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
The coefficient of x^(2^n+1)/(2^n+1) in A(x) for n>=1 begins:
[3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, ..., A293599(n), ...].

Links

Crossrefs

Cf. A293129, A293597, A293599.

Programs

  • Mathematica
    nmax = 80; Table[(CoefficientList[Series[Sum[x^((2*k - 1)^2)/((2*k - 1)*(1 - x^(2*k))^(2*k - 1)), {k, 1, 2*nmax + 1}], {x, 0, 2*nmax + 1}], x] * Range[0, 2*nmax + 1])[[2*n]], {n, 1, nmax}] (* Vaclav Kotesovec, Oct 15 2017 *)
  • PARI
    {a(n) = my(A, Ox = O(x^(2*n+1)));
A = sum(m=1, sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
(2*n-1)*polcoeff(A, 2*n-1)}
for(n=1, 80, print1(a(n), ", "))

A293599 The coefficient of x^(2^n+1)/(2^n+1) in the l.g.f. of A293598 for n>=1.

Original entry on oeis.org

3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, 3270933679995185, 344648907850020294305, 20381496562418327375031168210529, 303229033555187108276527297692992345985345, 533360801574481336406792124161160375221861972273961952144925889, 331572178130571824652402094592695034861147899073590997231695381294750188182312600193
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2017

Keywords

Comments

The l.g.f. of A293598 is Sum_{n>=1} x^((2*n-1)^2)/((2*n-1)*(1 - x^(2*n))^(2*n-1)).
The coefficient of x^(2^n+1)/(2^n+1) in the l.g.f. of A293597 equals 1 - a(n) for n>=2.
What is the rate of growth of this sequence?

Examples

			L.g.f. of A293598: Q(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
Explicitly,
Q(x) = x + (3)*x^3/3 + (5)*x^5/5 + 7*x^7/7 + (12)*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + (51)*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + (341)*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + (2925)*x^65/65 +...
This sequence equals the coefficient of x^(2^n+1)/(2^n+1) in Q(x) for n>=1.

Links

Crossrefs

Cf. A293129, A293597, A293598.

Programs

  • Mathematica
    nmax = 10; Table[(CoefficientList[Series[Sum[x^((2*k - 1)^2)/((2*k - 1)*(1 - x^(2*k))^(2*k - 1)), {k, 1, 2^nmax + 1}], {x, 0, 2^nmax + 1}], x] * Range[0, 2^nmax + 1])[[2^n + 2]], {n, 1, nmax}] (* Vaclav Kotesovec, Oct 15 2017 *)
  • PARI
    {A293598(n) = my(Q, Ox = O(x^(2*n+1)));
Q = sum(m=1, sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
(2*n-1)*polcoeff(Q, 2*n-1)}
for(n=0, 15, print1(A293598(2^n+1), ", "))
Showing 1-4 of 4 results.

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