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 1-6 of 6 results.

A249930 G.f. A(x) satisfies: 1+x = A(x)^2 + A(x)^6 - A(x)^7.

Original entry on oeis.org

1, 1, 5, 65, 1020, 17915, 336881, 6634391, 135085305, 2820748975, 60074608660, 1299896748546, 28496343865808, 631545920570632, 14126673078922850, 318514279669755470, 7231331734648562366, 165172699475903928921, 3793005825337785580290, 87518313783310367308770
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1020*x^4 + 17915*x^5 + ...
Related expansions.
A(x)^2 = 1 + 2*x + 11*x^2 + 140*x^3 + 2195*x^4 + 38520*x^5 + ...
A(x)^6 = 1 + 6*x + 45*x^2 + 560*x^3 + 8760*x^4 + 153546*x^5 + ...
A(x)^7 = 1 + 7*x + 56*x^2 + 700*x^3 + 10955*x^4 + 192066*x^5 + ...
where 1+x = A(x)^2 + A(x)^6 - A(x)^7.

Links

Crossrefs

Cf. A249926, A249927, A249928, A249929, A249931, A249932.

Programs

  • Maple
    S:= series(RootOf(1+x-(y^2+y^6-y^7),y,1),x,26):
seq(coeff(S,x,i),i=0..25); # Robert Israel, Mar 22 2023
  • PARI
    /* From 1+x = A(x)^2 + A(x)^6 - A(x)^7: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^2+Ser(A)^6-Ser(A)^7)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
  • PARI
    /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 5*x^2 - 15*x^3 - 20*x^4 - 15*x^5 - 6*x^6 - x^7 + x^2*O(x^n)));polcoeff(A,n)}
for(n=0, 25, print1(a(n) , ", "))

Formula

G.f.: 1 + Series_Reversion(x - 5*x^2 - 15*x^3 - 20*x^4 - 15*x^5 - 6*x^6 - x^7).

A249926 G.f. A(x) satisfies: 1+x = A(x)^2 + A(x)^4 - A(x)^5.

Original entry on oeis.org

1, 1, 3, 24, 229, 2449, 28035, 336100, 4165920, 52953884, 686517601, 9042628374, 120669757468, 1627932844657, 22166277534398, 304230231637560, 4204474770868230, 58458984141770754, 817176088436608178, 11477568712206346244, 161897000202383717334, 2292445680627209103645
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 229*x^4 + 2449*x^5 + 28035*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^4-1)/A(x)^17 + (A(x)-1)*(A(x)^4-1)*(A(x)^7-1)/A(x)^30 + (A(x)-1)*(A(x)^4-1)*(A(x)^7-1)*(A(x)^10-1)/A(x)^46 +
(A(x)-1)*(A(x)^4-1)*(A(x)^7-1)*(A(x)^10-1)*(A(x)^13-1)/A(x)^65 +...
Related expansions.
A(x)^2 = 1 + 2*x + 7*x^2 + 54*x^3 + 515*x^4 + 5500*x^5 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 136*x^3 + 1295*x^4 + 13816*x^5 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 190*x^3 + 1810*x^4 + 19316*x^5 +...
where 1+x = A(x)^2 + A(x)^4 - A(x)^5.

Crossrefs

Cf. A214692, A249927, A249928, A249929, A249930, A249931, A249932.

Programs

  • PARI
    /* From 1+x = A(x)^2 + A(x)^4 - A(x)^5: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^2+Ser(A)^4-Ser(A)^5)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
  • PARI
    /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 3*x^2 - 6*x^3 - 4*x^4 - x^5 + x^2*O(x^n)));polcoeff(A,n)}
for(n=0, 25, print1(a(n) , ", "))

Formula

G.f.: 1 + Series_Reversion(x - 3*x^2 - 6*x^3 - 4*x^4 - x^5).
G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(3*k-2)).
G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(n*(3*n-1)/2+6*n) * Product_{k=1..n} (A(x)^(3*k-2) - 1).

A249927 G.f. A(x) satisfies: 1+x = 2*A(x)^3 - A(x)^5.

Original entry on oeis.org

1, 1, 4, 40, 485, 6585, 95732, 1457636, 22947585, 370494965, 6101028934, 102074877086, 1730213141683, 29649526507055, 512810063004600, 8940267160930408, 156944360941491106, 2771866193105829798, 49218079130561578390, 878107603236732844610, 15733529061871743649380
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 40*x^3 + 485*x^4 + 6585*x^5 + 95732*x^6 +...
Related expansions.
A(x)^3 = 1 + 3*x + 15*x^2 + 145*x^3 + 1755*x^4 + 23793*x^5 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 290*x^3 + 3510*x^4 + 47586*x^5 +...
where 1+x = 2*A(x)^3 - A(x)^5.

Crossrefs

Cf. A249926, A249928, A249929, A249930, A249931, A249932.

Programs

  • PARI
    /* From 1+x = 2*A(x)^3 - A(x)^5: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(2*Ser(A)^3-Ser(A)^5)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
  • PARI
    /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 4*x^2 - 8*x^3 - 5*x^4 - x^5 + x^2*O(x^n)));polcoeff(A,n)}
for(n=0, 25, print1(a(n) , ", "))

Formula

G.f.: 1 + Series_Reversion(x - 4*x^2 - 8*x^3 - 5*x^4 - x^5).

A249928 G.f. A(x) satisfies: 1+x = A(x)^2 + A(x)^5 - A(x)^6.

Original entry on oeis.org

1, 1, 4, 42, 530, 7489, 113289, 1794784, 29397876, 493818165, 8460325159, 147264321834, 2596986868762, 46299572773077, 833111268412360, 15110637698827976, 275970530427257207, 5070759797283817869, 93671715612197557390, 1738657226550598494420, 32409645286487921390715
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 42*x^3 + 530*x^4 + 7489*x^5 + 113289*x^6 +...
Related expansions.
A(x)^2 = 1 + 2*x + 9*x^2 + 92*x^3 + 1160*x^4 + 16374*x^5 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 300*x^3 + 3775*x^4 + 53226*x^5 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 392*x^3 + 4935*x^4 + 69600*x^5 +...
where 1+x = A(x)^2 + A(x)^5 - A(x)^6.

Crossrefs

Cf. A249926, A249927, A249929, A249930, A249931, A249932.

Programs

  • PARI
    /* From 1+x = A(x)^2 + A(x)^5 - A(x)^6: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^2+Ser(A)^5-Ser(A)^6)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
  • PARI
    /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 4*x^2 - 10*x^3 - 10*x^4 - 5*x^5 - x^6 + x^2*O(x^n)));polcoeff(A,n)}
for(n=0, 25, print1(a(n) , ", "))

Formula

G.f.: 1 + Series_Reversion(x - 4*x^2 - 10*x^3 - 10*x^4 - 5*x^5 - x^6).

A249929 G.f. A(x) satisfies: 1+x = A(x)^3 + A(x)^4 - A(x)^6.

Original entry on oeis.org

1, 1, 6, 87, 1544, 30669, 652387, 14535220, 334846575, 7911062633, 190635605270, 4667362377021, 115772752576351, 2903222873038115, 73481220638826204, 1874686187998433232, 48159602555272931592, 1244712958804985611455, 32343121686417402278602, 844434166911645229308309
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 87*x^3 + 1544*x^4 + 30669*x^5 +...
Related expansions.
A(x)^3 = 1 + 3*x + 21*x^2 + 298*x^3 + 5280*x^4 + 104772*x^5 +...
A(x)^4 = 1 + 4*x + 30*x^2 + 424*x^3 + 7509*x^4 + 148968*x^5 +...
A(x)^6 = 1 + 6*x + 51*x^2 + 722*x^3 + 12789*x^4 + 253740*x^5 +...
where 1+x = A(x)^3 + A(x)^4 - A(x)^6.

Crossrefs

Cf. A249926, A249927, A249928, A249930, A249931, A249932.

Programs

  • PARI
    /* From 1+x = A(x)^3 + A(x)^4 - A(x)^6: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^3+Ser(A)^4-Ser(A)^6)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
  • PARI
    /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 6*x^2 - 15*x^3 - 14*x^4 - 6*x^5 - x^6 + x^2*O(x^n)));polcoeff(A,n)}
for(n=0, 25, print1(a(n) , ", "))

Formula

G.f.: 1 + Series_Reversion(x - 6*x^2 - 15*x^3 - 14*x^4 - 6*x^5 - x^6).

A249932 G.f. A(x) satisfies: 1+x = 2*A(x)^4 - A(x)^7.

Original entry on oeis.org

1, 1, 9, 189, 4893, 141771, 4399549, 143008579, 4806584226, 165685071354, 5825261477574, 208089042731934, 7531002619598458, 275548081670765638, 10175724606458649366, 378783118623851279250, 14197727780171976218883, 535401733845392629890255, 20298674469757725391376887
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 189*x^3 + 4893*x^4 + 141771*x^5 +...
Related expansions.
A(x)^4 = 1 + 4*x + 42*x^2 + 868*x^3 + 22435*x^4 + 649488*x^5 +...
A(x)^7 = 1 + 7*x + 84*x^2 + 1736*x^3 + 44870*x^4 + 1298976*x^5 +...
where 1+x = 2*A(x)^4 - A(x)^7.

Crossrefs

Cf. A249926, A249927, A249928, A249929, A249930, A249931.

Programs

  • PARI
    /* From 1+x = 2*A(x)^4 - A(x)^7: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(2*Ser(A)^4-Ser(A)^7)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
  • PARI
    /* From Series Reversion: */
{a(n)=local(A=1+serreverse(x - 9*x^2 - 27*x^3 - 33*x^4 - 21*x^5 - 7*x^6 - x^7 + x^2*O(x^n)));polcoeff(A,n)}
for(n=0, 25, print1(a(n) , ", "))

Formula

G.f.: 1 + Series_Reversion(x - 9*x^2 - 27*x^3 - 33*x^4 - 21*x^5 - 7*x^6 - x^7).
Showing 1-6 of 6 results.

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