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 1-5 of 5 results.

A268302 G.f.: Sum_{n>=1} x^(n*(n-1)/2) * (G(x)^n + 1/G(x)^(n-1)), where G(x) is the g.f. of A268300.

Original entry on oeis.org

3, 8, 28, 144, 736, 4024, 22912, 134784, 813476, 5010904, 31379808, 199196320, 1278911808, 8290414024, 54186864896, 356711621984, 2362968349568, 15739688709864, 105357470567228, 708338644347808, 4781146692837856, 32387329985982176, 220104493513881920, 1500273861724289984, 10253983269166864256, 70258772726034956688, 482514972838806347776, 3320848006096569464080, 22900703924095461843008, 158216154716853989543080
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2016

Keywords

Examples

			G.f.: A(x) = 3 + 8*x + 28*x^2 + 144*x^3 + 736*x^4 + 4024*x^5 + 22912*x^6 + 134784*x^7 + 813476*x^8 + 5010904*x^9 + 31379808*x^10 +...
such that
A(x) = Sum_{n>=1} x^(n*(n-1)/2) * (G(x)^n + 1/G(x)^(n-1)),
that is,
A(x) = (G(x) + 1) + x*(G(x)^2 + 1/G(x)) + x^3*(G(x)^3 + 1/G(x)^2) + x^6*(G(x)^4 + 1/G(x)^3) + x^10*(G(x)^5 + 1/G(x)^4) + x^15*(G(x)^6 + 1/G(x)^5) +...,
where
G(x) = 2 + 7*2*x/4 + 119*2*x^2/4^2 + 2118*2*x^3/4^3 + 42523*2*x^4/4^4 + 914922*2*x^5/4^5 + 20745494*2*x^6/4^6 + 487390092*2*x^7/4^7 + 11764545555*2*x^8/4^8 + 289962708802*2*x^9/4^9 +...+ A268300(n)*2*x^n/4^n +...
satisfies:
-1 = Product_{n>=1} (1-x^n) * (1 - x^n/G(x)) * (1 - x^(n-1)*G(x)).

Crossrefs

Cf. A268300, A268301.

Formula

G.f.: Product_{n>=1} (1-x^n) * (1 + x^n/G(x)) * (1 + x^(n-1)*G(x)), where G(x) is the g.f. of A268300.

A268299 G.f. A(x) satisfies: -1 = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n/x) * (1 - A(x)^(n-1)*x).

Original entry on oeis.org

2, 7, 84, 1240, 20942, 382344, 7354688, 146810440, 3012778758, 63167322872, 1347251937632, 29138746861200, 637584335364362, 14088532800477752, 313936020646727040, 7046500093908958288, 159171390375064583380, 3615669944253537267048, 82541551931101193203004, 1892725670848222011475776, 43575217427267416453289838, 1006843304895182755611475824, 23340548167572913996786290328
Offset: 1

Views

Author

Paul D. Hanna, Feb 26 2016

Keywords

Comments

The g.f. utilizes the Jacobi Triple Product: Product_{n>=1} (1-x^n)*(1 - x^n/a)*(1 - x^(n-1)*a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.
Equals the row sums of triangle A354650. - Paul D. Hanna, Jul 27 2022

Examples

			G.f.: A(x) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...
where A(x) satisfies the Jacobi Triple Product:
-1 = (1-A(x))*(1-A(x)/x)*(1-x) * (1-A(x)^2)*(1-A(x)^2/x)*(1-A(x)*x) * (1-A(x)^3)*(1-A(x)^3/x)*(1-A(x)^2*x) * (1-A(x)^4)*(1-A(x)^4/x)*(1-A(x)^3*x) * (1-A(x)^5)*(1-A(x)^5/x)*(1-A(x)^4*x) * (1-A(x)^6)*(1-A(x)^6/x)*(1-A(x)^5*x) +...
also
1/x = (1-A(x))*(1-A(x)*x)*(1-1/x) * (1-A(x)^2)*(1-A(x)^2*x)*(1-A(x)/x) * (1-A(x)^3)*(1-A(x)^3*x)*(1-A(x)^2/x) * (1-A(x)^4)*(1-A(x)^4*x)*(1-A(x)^3/x) * (1-A(x)^5)*(1-A(x)^5*x)*(1-A(x)^4/x) * (1-A(x)^6)*(1-A(x)^6*x)*(1-A(x)^5/x) *...
further,
-1 = (1-x) - A(x)*(1-x^3)/x + A(x)^3*(1-x^5)/x^2 - A(x)^6*(1-x^7)/x^3 + A(x)^10*(1-x^9)/x^4 - A(x)^15*(1-x^11)/x^5 + A(x)^21*(1-x^13)/x^6 +...
RELATED SERIES.
The series reversion of g.f. A(x) equals x*Q(x), where Q(x) begins:
Q(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 - 1979248977748/2*x^10/4^10 +...+ A268301(n)/2*x^n/4^n +...
and where Q(x) satisfies the Jacobi Triple Product:
-1 = (1-x)*(1-x*Q(x))*(1-1/Q(x)) * (1-x^2)*(1-x^2*Q(x))*(1-x/Q(x)) * (1-x^3)*(1-x^3*Q(x))*(1-x^2/Q(x)) * (1-x^4)*(1-x^4*Q(x))*(1-x^3/Q(x)) * (1-x^5)*(1-x^5*Q(x))*(1-x^4/Q(x)) * (1-x^6)*(1-x^6*Q(x))*(1-x^5/Q(x)) *...

Links

Crossrefs

Cf. A354650, A268300, A268301, A268650.

Programs

  • Mathematica
    (* Calculation of constant d: *) 1/r /. FindRoot[{r*QPochhammer[1/r, s]*QPochhammer[r, s]* QPochhammer[s, s] == 1 - r, (Log[1 - s] + QPolyGamma[0, 1, s])/(s*Log[s]) - Derivative[0, 1][QPochhammer][1/r, s]/QPochhammer[1/r, s] - Derivative[0, 1][QPochhammer][r, s]/QPochhammer[r, s] - Derivative[0, 1][QPochhammer][s, s]/ QPochhammer[s, s] == 0}, {r, 1/24}, {s, 1/8}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)
  • PARI
    {a(n) = my(Q=1/2, t=floor(sqrt(2*n+1)+1/2)); for(i=0, n, Q = (Q + sum(m=-t, t, x^(m*(m-1)/2) * (-Q)^m +x*O(x^n)) )/2 ); polcoeff(serreverse(x*Q), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) -1 = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-x)^n.
(2) -1 = Sum_{n>=0} A(x)^(n*(n+1)/2) * (1 - x^(2*n+1)) / (-x)^n.
(3) 1/x = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-1/x)^n.
(4) 1/x = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n*x) * (1 - A(x)^(n-1)/x).
(5) A(x) = Series_Reversion( x*Q(x) ), where Q(x) is the g.f. of A268301 and satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n*Q(x)) * (1 - x^(n-1)/Q(x)).
(6) x = Sum_{n>=1} a(n) * x^n * Q(x)^n, where Q(x) = Sum_{n>=0} A268301(n)/2*(x/4)^n.
a(n) ~ c * d^n / n^(3/2), where d = 24.827421130209954998234265953843191542003179657... and c = 0.020386712793003585674903530668163000681070027... . - Vaclav Kotesovec, Mar 02 2016

A268301 G.f. satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where g.f. A(x) = Sum_{n>=0} a(n)/2*(x/4)^n.

Original entry on oeis.org

1, -7, -70, -795, -13802, -277782, -6093708, -139376659, -3297234754, -79988099074, -1979248977748, -49758116194846, -1267321717299236, -32631825106297228, -848030793254951704, -22214311484843607811, -585938143786366837938, -15548874443787002057610, -414829266882771282611204, -11120089118043870668697578, -299364678394845043715844268, -8090271856987498430846360564
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2016

Keywords

Comments

The g.f. utilizes the Jacobi Triple Product: Product_{n>=1} (1-x^n)*(1 - x^n*a)*(1 - x^(n-1)/a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * a^n.

Examples

			G.f.: A(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 - 1979248977748/2*x^10/4^10 -...
where g.f. A(x) satisfies the Jacobi Triple Product:
-1 = (1-x)*(1-x*A(x))*(1-1/A(x)) * (1-x^2)*(1-x^2*A(x))*(1-x/A(x)) * (1-x^3)*(1-x^3*A(x))*(1-x^2/A(x)) * (1-x^4)*(1-x^4*A(x))*(1-x^3/A(x)) * (1-x^5)*(1-x^5*A(x))*(1-x^4/A(x)) * (1-x^6)*(1-x^6*A(x))*(1-x^5/A(x)) *...
also
A(x) = (1-x)*(1-x/A(x))*(1-A(x)) * (1-x^2)*(1-x^2/A(x))*(1-x*A(x)) * (1-x^3)*(1-x^3/A(x))*(1-x^2*A(x)) * (1-x^4)*(1-x^4/A(x))*(1-x^3*A(x)) * (1-x^5)*(1-x^5/A(x))*(1-x^4*A(x)) * (1-x^6)*(1-x^6/A(x))*(1-x^5*A(x)) *...
RELATED SERIES.
1/A(x) = 2 + 7*2*x/4 + 119*2*x^2/4^2 + 2118*2*x^3/4^3 + 42523*2*x^4/4^4 + 914922*2*x^5/4^5 + 20745494*2*x^6/4^6 + 487390092*2*x^7/4^7 + 11764545555*2*x^8/4^8 + 289962708802*2*x^9/4^9 +...+ A268300(n)*2*x^n/4^n +...
Series_Reversion( x*A(x) ) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...+ A268299(n)*x^n +..., an integer series.
Let J(x) = Sum_{n>=1} x^(n*(n-1)/2) * (A(x)^(n-1) + 1/A(x)^n),
then J(x) is an integer series:
J(x) = 3 + 8*x + 28*x^2 + 144*x^3 + 736*x^4 + 4024*x^5 + 22912*x^6 + 134784*x^7 + 813476*x^8 + 5010904*x^9 + 31379808*x^10 +..+ A268302(n)*x^n +...
and J(x) = Product_{n>=1} (1-x^n) * (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)).
Conjecture: Product_{n>=1} (1-x^n) * (1 + k*x^n*A(x)) * (1 + k*x^(n-1)/A(x)) yields an integer series for all integer k.

Links

Crossrefs

Cf. A268300, A268302, A268299, A190791.

Programs

  • PARI
    {a(n) = my(A=1/2+x,t=floor(sqrt(2*n+1)+1/2)); for(i=0,n, A = (A + sum(m=-t,t, x^(m*(m-1)/2) * (-A)^m +x*O(x^n)) )/2 ); 2*4^n * polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

Given g.f. A(x) = Sum_{n>=0} a(n)/2 * (x/4)^n, then g.f. also satisfies:
(1) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n,
(2) A(x) = Product_{n>=1} (1-x^n) * (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)),
(3) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
(4) x = Sum_{n>=1} A268299(n) * x^n * A(x)^n.
a(n) is odd iff n = 2^k-1 for k>=0 (conjecture).
a(n) ~ -c * d^n / n^(3/2), where d = 29.101591090693617170487962339050658... and c = 0.1385938593465955724446602611055779... . - Vaclav Kotesovec, Mar 02 2016

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

Original entry on oeis.org

1, 0, 3, -3, 1, 0, 9, -18, 21, -15, 6, -1, 0, 22, -56, 116, -182, 196, -140, 64, -17, 2, 0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5, 0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14, 0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2022

Keywords

Comments

Row sums equal A000108, the Catalan numbers:
Sum_{k=0..3*n} T(n,k) = A000108(n) for n >= 0.
T(n,3*n) = (-1)^(n-1) * A000108(n-1) for n >= 1 (Catalan numbers).
Conjecture: T(n,1) = A000716(n) for n >= 1 (number of partitions of n into parts of 3 kinds).
The generating functions of some related sequences are given as follows.
(1) A(x,x) = Sum_{n>=0} A355351(n)*x^n.
(2) A(x,2*x) = Sum_{n>=0} A355352(n)*x^n.
(3) A(x,3*x) = Sum_{n>=0} A355353(n)*x^n.
(4) A(x,4*x) = Sum_{n>=0} A355354(n)*x^n.
(5) A(x,5*x) = Sum_{n>=0} A355355(n)*x^n.
(6) A(x,x^2) = Sum_{n>=0} A355356(n)*x^n.
(7) A(x^2,x) = Sum_{n>=0} A355357(n)*x^n.
(8) A(x,x*y) = Sum_{n>=0} x^n * Sum_{k=0..n} A355350(n,k) * y^k.
(9) 1/A(4*x,-1) = 2*Sum_{n>=0} A268300(n)*x^n.
(10) A(x,2) = -Sum_{n>=0} A355871(n)*x^n.
SPECIFIC VALUES.
(V.1) A(x,y) = -exp(-Pi) at x = exp(-2*Pi) and y = exp(Pi) * Pi^(1/4)/gamma(3/4).
(V.2) A(x,y) = -exp(-2*Pi) at x = exp(-4*Pi) and y = exp(2*Pi) * Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) A(x,y) = -exp(-3*Pi) at x = exp(-6*Pi) and y = exp(3*Pi) * Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) A(x,y) = -exp(-4*Pi) at x = exp(-8*Pi) and y = exp(4*Pi) * Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) A(x,y) = -exp(-sqrt(3)*Pi) at x = exp(-2*sqrt(3)*Pi) and y = exp(sqrt(3)*Pi) * gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

Examples

			G.f.: A(x,y) = 1/(1-y) + x*(y^3 - 3*y^2 + 3*y)/(1-y)^3 + x^2*(-y^6 + 6*y^5 - 15*y^4 + 21*y^3 - 18*y^2 + 9*y)/(1-y)^5 + x^3*(2*y^9 - 17*y^8 + 64*y^7 - 140*y^6 + 196*y^5 - 182*y^4 + 116*y^3 - 56*y^2 + 22*y)/(1-y)^7 + x^4*(-5*y^12 + 57*y^11 - 297*y^10 + 934*y^9 - 1971*y^8 + 2934*y^7 - 3148*y^6 + 2436*y^5 - 1329*y^4 + 496*y^3 - 144*y^2 + 51*y)/(1-y)^9 + x^5*(14*y^15 - 200*y^14 + 1330*y^13 - 5455*y^12 + 15412*y^11 - 31724*y^10 + 49060*y^9 - 57914*y^8 + 52462*y^7 - 36302*y^6 + 18846*y^5 - 7005*y^4 + 1680*y^3 - 270*y^2 + 108*y)/(1-y)^11 + ...
where
y = ... + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also,
y = (1 - x*A(x,y))*(1 - 1/A(x,y))*(1-x) * (1 - x^2*A(x,y))*(1 - x/A(x,y))*(1-x^2) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y))*(1-x^3) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y))*(1-x^4) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y))*(1-x^n) * ...
This triangle of coefficients T(n,k) of x^n*y^k/(1-y)^(2*n+1) in A(x,y), for k = 0..3*n in row n, begins
n = 0: [1];
n = 1: [0, 3, -3, 1];
n = 2: [0, 9, -18, 21, -15, 6, -1];
n = 3: [0, 22, -56, 116, -182, 196, -140, 64, -17, 2];
n = 4: [0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5];
n = 5: [0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14];
n = 6: [0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42];
n = 7: [0, 429, -63, 18281, -134985, 594399, -1941037, 4947447, -10062669, 16571700, -22316250, 24716922, -22564425, 16956135, -10435305, 5210319, -2078910, 647565, -151825, 25215, -2646, 132]; ...
The rightmost border equals the signed Catalan numbers (A000108) shifted right one place.
Column 1 appears to equal A000716 (ignoring the initial term).
Example: at y = x, we have the g.f. of A355351:
A(x,x) = 1/(1-x) + x*(3*x - 3*x^2 + x^3)/(1-x)^3 + x^2*(9*x - 18*x^2 + 21*x^3 - 15*x^4 + 6*x^5 - x^6)/(1-x)^5 + x^3*(22*x - 56*x^2 + 116*x^3 - 182*x^4 + 196*x^5 - 140*x^6 + 64*x^7 - 17*x^8 + 2*x^9)/(1-x)^7 + ... = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + ... + A355351(n)*x^n + ...
where x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,x)^n.

Links

Crossrefs

Cf. A000108 (row sums), A355871 (y=2).
Cf. A355350 (related triangle), A355351 (y=x), A355352 (y=2*x), A355353 (y=3*x), A355354 (y=4*x), A355355 (y=5*x), A355356 (y=x^2), A355357 (x=x^2,y=x).
Cf. A355360 (related triangle), A000716.

Programs

  • PARI
    {T(n,k) = my(A=[1/(1-y)],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( (y - sum(m=-t,t, (-1)^m * x^(m*(m+1)/2) * Ser(A)^m )), #A-1,x)/(1-y)^2);polcoeff(A[n+1]*(1-y)^(2*n+1),k,y)}
for(n=0,12, for(k=0,3*n, print1( T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) y = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^n), by the Jacobi triple product identity.

A349031 The "Fouriest" numbers: numbers that can be expressed as a string of 4's longer than the original number in some base.

Original entry on oeis.org

624, 3124, 6220, 15624, 37324, 78124, 78432, 223948, 390624, 549028, 1343692, 1953124, 3843200, 8062156, 9586980, 9765624, 26902404, 48372940, 48828124, 76695844, 188316832, 244140624, 290237644, 613566756, 1220703124, 1318217828, 1741425868, 4908534052
Offset: 1

Views

Author

David Consiglio, Jr., Nov 06 2021

Keywords

Comments

Inspired by Saturday Morning Breakfast Cereal comics.
The "Fouriest" numbers. The number shown in the comment, 624, is correctly identified as a term of the sequence.

Examples

			624 is a member of this sequence because 624 expressed in base 5 is 4444. 4444 has 4 digits and 624 has only 3.

Links

Crossrefs

For the Fouriest transform see A268236-A268238, A268300. - N. J. A. Sloane, Nov 19 2021

Programs

  • Python
    from math import log
def A349031(limit):
    super_fours = []
    for length in range(4,int(log(limit,5))+1):
        fours = "4"*length
        for base in range(5, 10):
            keep = 4*(1-base**length)//(1-base)
            if len(str(keep)) < len(fours) and keep < limit:
                super_fours.append(keep)
    return sorted(super_fours)
result = A349031(10**20)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.