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.

A157303 G.f. A(x) satisfies the condition that both G(x) = A(x/G(x)) and H(x) = A(x*H(x)) have zeros for every other coefficient after initial terms; g.f. of dual sequence A157306 satisfies the same condition.

Original entry on oeis.org

1, 1, 1, -4, -19, 134, 1074, -10158, -110067, 1302086, 17451662, -248857456, -3948994550, 66104803660, 1210719480268, -23304935437410, -484165206834051, 10541640152174406, 245149591293286518, -5958110923842801192
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 - 4*x^3 - 19*x^4 + 134*x^5 + 1074*x^6 --++...
...
Let G(x) = A(x/G(x)) so that A(x) = G(x*A(x)) then
G(x) = 1 + x - 5*x^3 + 183*x^5 - 14352*x^7 + 1857199*x^9 -+...
has alternating zeros in the coefficients (cf. A157302):
[1,1,0,-5,0,183,0,-14352,0,1857199,0,-355082433,0,94134281460,0,...]
...
Let H(x) = A(x*H(x)) so that A(x) = H(x/A(x)) then
H(x) = 1 + x + 2*x^2 - 26*x^4 + 1378*x^6 - 141202*x^8 +-...
has alternating zeros in the coefficients (cf. A157304):
[1,1,2,0,-26,0,1378,0,-141202,0,22716418,0,-5218302090,0,...].
...
ZERO COEFFICIENTS IN POWERS OF G.F. A(x).
Even powers A(x)^(2n) yield zeros at odd positions 2n-1 for n>=2:
A^4: [1,4,10, 0, -105,228,5442,-24048,-535293,3588828,...];
A^6: [1,6,21,26,-144, 0, 8415,-17802,-849771,3596382,...];
A^8: [1,8,36,80,-110,-384,10608, 0, -1143105,2366472,...];
A^10:[1,10,55,170,65,-718,11580,26520,-1381080, 0, ...];
...
Odd negative powers 1/A(x)^(2n-1) yield zeros at even positions 2n for n>=1:
A^-1: [1,-1, 0, 5,10,-168,-762,12297,87222,-1524622,...];
A^-3: [1,-3,3,14, 0, -549,-1173,41184,180558,-5088382,...];
A^-5: [1,-5,10,15,-45,-891, 0, 71760,152670,-9042640,...];
A^-7: [1,-7,21,0,-105,-1092,2604,99042, 0, -13000393,...]; ...

Crossrefs

Cf. A157302, A157304, A157305, A157306 (dual), A157307.

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(x^2/serreverse(x*Ser(A)))[ #A], t, 0))); A[n+1]}

Formula

For n>=2, [x^(2n-1)] A(x)^(2n) = 0.
For n>=1, [x^(2n)] 1/A(x)^(2n-1) = 0.
G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = x/Series_Reversion(x*A(x)) = g.f. of A157302.
G.f. satisfies: A(x) = H(x/A(x)) where H(x) = A(x*H(x)) = Series_Reversion(x/A(x))/x = g.f. of A157304.

A157304 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x/F(x)^2) have zeros for every other coefficient after initial terms; g.f. of dual sequence A157307 satisfies the same condition.

Original entry on oeis.org

1, 1, 2, 0, -26, 0, 1378, 0, -141202, 0, 22716418, 0, -5218302090, 0, 1619288968386, 0, -653379470919714, 0, 333014944014777730, 0, -209463165121436380282, 0, 159492000935562428176162, 0, -144654795258284936534929586, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2009

Keywords

Comments

After initial 2 terms, reversing signs yields the complementary sequence A157305, which has very similar properties.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 - 26*x^4 + 1378*x^6 - 141202*x^8 +-...
...
Let F(x) = A(x/F(x)^2) so that A(x) = F(x*A(x)^2) then
F(x) = 1 + x - 5*x^3 + 183*x^5 - 14352*x^7 + 1857199*x^9 -+...
has alternating zeros in the coefficients (cf. A157302):
[1,1,0,-5,0,183,0,-14352,0,1857199,0,-355082433,0,94134281460,0,...].
...
COEFFICIENTS IN ODD NEGATIVE POWERS OF G.F. A(x).
A^1 : [(1), 1,2,0,-26,0,1378,0,-141202,0,22716418,...];
A^-1: [1,(-1),-1,3,25,-57,-1397,2967,143057,...];
A^-3: [1,-3,(0),14,57,-333,-3880,18036,415665,...];
A^-5: [1,-5,5,(25),50,-766,-5370,44370,637275,...];
A^-7: [1,-7,14,28,(0),-1246,-5334,79148,770469,...];
A^-9: [1,-9,27,15,-81,(-1647),-3519,117981,784998,...];
A^-11:[1,-11,44,-22,-165,-1859,(0),155584,662046,...];
A^-13:[1,-13,65,-91,-208,-1820,4836,(186576),396942,...];
A^-15:[1,-15,90,-200,-150,-1548,10370,206280,(0),...];
A^-17:[1,-17,119,-357,85,-1173,15895,211395,-504577,(-31572383),...];
...
When scaled, the coefficients shown above in parenthesis
forms the coefficients of the function F(x) = A(x/F(x)^2):
F: [1,-1/(-1),0,25/(-5),0,-1647/(-9),0,186576/(-13),0,-31572383/(-17),...].

Crossrefs

Cf. A157302, A157303, A157305 (complement), A157306, A157307 (dual).

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(serreverse(x/Ser(A))/x)[n+1]}

Formula

For n>=1, [x^(2n)] 1/A(x)^(4n-1) = 0.
G.f. satisfies: A(x) = F(x*A(x)^2) where F(x) = A(x/F(x)^2) = sqrt(x/Series_Reversion(x*A(x)^2)) = g.f. of A157302.
G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = x/Series_Reversion(x*A(x)) = g.f. of A157303.

A157305 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)^2) have zeros for every other coefficient after initial terms; dual sequence A157302 satisfies the same condition.

Original entry on oeis.org

1, 1, -2, 0, 26, 0, -1378, 0, 141202, 0, -22716418, 0, 5218302090, 0, -1619288968386, 0, 653379470919714, 0, -333014944014777730, 0, 209463165121436380282, 0, -159492000935562428176162, 0, 144654795258284936534929586, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - 2*x^2 + 26*x^4 - 1378*x^6 + 141202*x^8 -+...
...
Let F(x) = A(x*F(x)^2) so that A(x) = F(x/A(x)^2) then
F(x) = 1 + x - 7*x^3 + 242*x^5 - 17771*x^7 + 2189294*x^9 -+...
has alternating zeros in the coefficients (cf. A157304):
[1,1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,104785114020,0,...].
...
COEFFICIENTS IN ODD POWERS OF G.F. A(x).
A^1: [(1),1,-2,0,26,0,-1378,0,141202,0,-22716418,0,...];
A^3: [1,(3),-3,-11,84,168,-4376,-8580,438348,865776,...];
A^5: [1,5,(0),-30,115,601,-7120,-30280,726680,2987400,...];
A^7: [1,7,7,(-49),91,1253,-8743,-65519,964768,6410880,...];
A^9: [1,9,18,-60,(0),1998,-8418,-112284,1106775,11070241,...];
A^11:[1,11,33,-55,-154,(2662),-5566,-166034,1108657,...];
A^13:[1,13,52,-26,-351,3055,(0),-220116,935051,23169939,...];
A^15:[1,15,75,35,-555,3003,7995,(-266565),565635,29818365,...];
A^17:[1,17,102,136,-714,2380,17646,-297160,(0),36161142,...];
A^19:[1,19,133,285,-760,1140,27740,-304608,-739670,(41596586),...];
...
When scaled, the coefficients shown above in parenthesis
forms the coefficients of the function F(x) = A(x*F(x)^2):
F: [1,3/3,0,-49/7,0,2662/11,0,-266565/15,0,41596586/19,0,...].

Crossrefs

Cf. A157306, A157307, A157302 (dual), A157303, A157304.

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(x/serreverse(x*Ser(A)))[n+1]}

Formula

For n>=2, [x^(2n)] A(x)^(4n+1) = 0.
G.f. satisfies: A(x) = F(x/A(x)^2) where F(x) = A(x*F(x)^2) = sqrt(Series_Reversion(x/A(x)^2)/x) = g.f. of A157307.
G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = Series_Reversion(x/A(x))/x = g.f. of A157306.

A157307 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x/F(x)^2) have zeros for every other coefficient after initial terms; g.f. of dual sequence A157304 satisfies the same condition.

Original entry on oeis.org

1, 1, 0, -7, 0, 242, 0, -17771, 0, 2189294, 0, -404590470, 0, 104785114020, 0, -36267349272243, 0, 16185534555969878, 0, -9056084951164614194, 0, 6210575966204050489916, 0, -5124228589785627978616702, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - 7*x^3 + 242*x^5 - 17771*x^7 + 2189294*x^9 -+...
...
Let F(x) = A(x/F(x)^2) so that A(x) = F(x*A(x)^2) then
F(x) = 1 + x - 2*x^2 + 26*x^4 - 1378*x^6 + 141202*x^8 -+...
has alternating zeros in the coefficients (cf. A157305):
[1,1,-2,0,26,0,-1378,0,141202,0,-22716418,0,5218302090,0,...].
...
COEFFICIENTS IN ODD NEGATIVE POWERS OF G.F. A(x).
A^1 : [(1),1,0,-7,0,242,0,-17771,0,2189294,0,-404590470,0,...];
A^-1: [1,(-1),1,6,-13,-222,506,16932,-37709,-2127126,4595294,...];
A^-3: [1,-3,(6),11,-69,-537,2806,45282,-215781,-5963673,...];
A^-5: [1,-5,15,(0),-140,-601,6245,62380,-503935,-8911515,...];
A^-7: [1,-7,28,-35,(-182),-392,9968,65519,-860825,-10670499,...];
A^-9: [1,-9,45,-102,-135,(0),13128,54504,-1240416,-11070241,...];
A^-11:[1,-11,66,-209,77,341,(15158),31460,-1598696,-10074240,...];
A^-13:[1,-13,91,-364,546,221,16107,(0),-1899508,-7767240,...];
A^-15:[1,-15,120,-575,1380,-978,17040,-36375,(-2118030),...];
A^-17:[1,-17,153,-850,2703,-4114,20502,-76772,-2240175,(0),...];
...
When scaled, the coefficients shown above in parenthesis
forms the coefficients of the function F(x) = A(x/F(x)^2):
F: [1,-1/(-1),6/(-3),0,-182/(-7),0,15158/(-11),0,-2118030/(-15),0,...].

Crossrefs

Cf. A157305, A157306, A157302, A157303, A157304 (dual).

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(serreverse(x/Ser(A))/x)[n+1]}

Formula

For n>=1, [x^(2n+1)] 1/A(x)^(4n+1) = 0.
G.f. satisfies: A(x) = F(x*A(x)^2) where F(x) = A(x/F(x)^2) = sqrt(x/Series_Reversion(x*A(x)^2)) = g.f. of A157305.
G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = x/Series_Reversion(x*A(x)) = g.f. of A157306.

A157302 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)^2) have zeros for every other coefficient after initial terms; g.f. of dual sequence A157305 satisfies the same condition.

Original entry on oeis.org

1, 1, 0, -5, 0, 183, 0, -14352, 0, 1857199, 0, -355082433, 0, 94134281460, 0, -33120720127500, 0, 14959943533260783, 0, -8447188671812872887, 0, 5834800994047642310223, 0, -4842259038722174600622240, 0
Offset: 0

Views

Author

Paul D. Hanna, Feb 28 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - 5*x^3 + 183*x^5 - 14352*x^7 + 1857199*x^9 -+...
...
Let F(x) = A(x*F(x)^2) so that A(x) = F(x/A(x)^2) then
F(x) = 1 + x + 2*x^2 - 26*x^4 + 1378*x^6 - 141202*x^8 +-...
has alternating zeros in the coefficients (cf. A157304):
[1,1,2,0,-26,0,1378,0,-141202,0,22716418,0,-5218302090,0,...].
...
COEFFICIENTS IN ODD POWERS OF G.F. A(x).
A^1: [(1),1,0,-5,0,183,0,-14352,0,1857199,0,...];
A^3: [1,(3),3,-14,-30,534,1173,-42432,-91602,5522926,...];
A^5: [1,5,(10),-15,-95,766,3810,-65545,-300930,8800450,...];
A^7: [1,7,21,(0),-175,777,7518,-79148,-610554,11321338,...];
A^9: [1,9,36,39,(-234),513,11640,-79866,-990603,...];
A^11:[1,11,55,110,-220,(0),15367,-66132,-1402005,...];
A^13:[1,13,78,221,-65,-624,(17914),-38571,-1801215,...];
A^15:[1,15,105,380,315,-1077,18760,(0),-2145855,...];
A^17:[1,17,136,595,1020,-901,17952,45084,(-2400434),...];
A^19:[1,19,171,874,2166,570,16473,91656,-2541060,(0),...];
...
When scaled, the coefficients shown above in parenthesis
forms the coefficients of the function F(x) = A(x*F(x)^2):
F: [1,3/3,10/5,0,-234/9,0,17914/13,0,-2400434/17,0,...].

Crossrefs

Cf. A157303, A157304, A157305 (dual), A157306, A157307.

Programs

  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(x/serreverse(x*Ser(A)))[n+1]}

Formula

For n>=2, [x^(2n-1)] A(x)^(4n-1) = 0.
G.f. satisfies: A(x) = F(x/A(x)^2) where F(x) = A(x*F(x)^2) = sqrt(Series_Reversion(x/A(x)^2)/x) = g.f. of A157304.
G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = Series_Reversion(x/A(x))/x = g.f. of A157303.
Showing 1-5 of 5 results.

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