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.

A143045 G.f.: A(x) = x + A(-x)^2.

Original entry on oeis.org

1, 1, -2, -3, 10, 18, -68, -131, 530, 1062, -4476, -9198, 39844, 83332, -368136, -780003, 3497058, 7483806, -33940940, -73210874, 335103340, 727473084, -3355045304, -7322240718, 33982884884, 74498594492, -347600543192, -764936992764, 3585459509640, 7916276980872, -37253166379536
Offset: 1

Views

Author

Paul D. Hanna, Jul 19 2008, Jul 20 2008

Keywords

Examples

			A(x) = x + x^2 - 2*x^3 - 3*x^4 + 10*x^5 + 18*x^6 - 68*x^7 - 131*x^8 +...
A(x)^2 = x^2 + 2*x^3 - 3*x^4 - 10*x^5 + 18*x^6 + 68*x^7 - 131*x^8 - 530*x^9 +...
A(x)^3 = x^3 + 3*x^4 - 3*x^5 - 20*x^6 + 18*x^7 + 153*x^8 - 131*x^9 -++-...
Note that a bisection of A^3 equals a bisection of A.

Links

Crossrefs

Cf. A143046, A143048, A143049, A213252, A100238 (related by series reversion).

Programs

  • Mathematica
    Rest[CoefficientList[1 + InverseSeries[Series[(Sqrt[1+4*x-4*x^2]-1)/2 + x^2, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Dec 27 2013 *)
  • PARI
    {a(n)=if(n<1,0,polcoeff(serreverse((sqrt(1+4*x-4*x^2 +x*O(x^n))-1)/2 + x^2),n))}
  • PARI
    {a(n)=local(A=x+x^2);for(i=0,n,A=x+subst(A,x,(-x+x*O(x^n)))^2);polcoeff(A,n)}
  • PARI
    {a(n)=local(A=x+x*O(x^n));for(i=0,n,A = x + x^2 - 2*x*A^2 + A^4);polcoeff(A,n)}

Formula

G.f.: A(x) = Series_Reversion( (sqrt(1+4*x-4*x^2)-1)/2 + x^2 ).
G.f. satisfies: A(x) = x + ( x - A(x)^2 )^2.
G.f. satisfies: [A(x)^3 - A(-x)^3]/2 = x*[A(x) + A(-x)]/2.
Recurrence: 27*(n-2)*(n-1)*n*(16*n^3 - 160*n^2 + 521*n - 552)*a(n) = 18*(n-2)*(n-1)*(32*n - 105)*a(n-1) - 12*(n-2)*(384*n^5 - 5376*n^4 + 29288*n^3 - 77560*n^2 + 99709*n - 49665)*a(n-2) + 48*(2*n - 7)*(32*n^3 - 248*n^2 + 623*n - 505)*a(n-3) + 64*(n-4)*(2*n - 9)*(2*n - 7)*(16*n^3 - 112*n^2 + 249*n - 175)*a(n-4). - Vaclav Kotesovec, Dec 27 2013
Limit n->infinity |a(n)|^(1/n) = 4/3*sqrt(3+2*sqrt(3)) = 3.3899463424498833... - Vaclav Kotesovec, Dec 27 2013

A143046 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^3.

Original entry on oeis.org

1, 1, -3, -6, 35, 87, -588, -1578, 11511, 32223, -245883, -706824, 5556564, 16267508, -130617600, -387533058, 3161190783, 9474886287, -78241316361, -236394953670, 1971270824859, 5994591989967, -50388913722480, -154052058035736
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			G.f.: A(x) = 1 + x - 3*x^2 - 6*x^3 + 35*x^4 + 87*x^5 - 588*x^6 - 1578*x^7 +...
where
A(x)^3 = 1 + 3*x - 6*x^2 - 35*x^3 + 87*x^4 + 588*x^5 - 1578*x^6 - 11511*x^7 +...
A(x)^4 = 1 + 4*x - 6*x^2 - 56*x^3 + 87*x^4 + 1008*x^5 - 1578*x^6 - 20464*x^7 +...
Note that a bisection of A^4 equals a bisection of A^3.

Links

Crossrefs

Cf. A143045, A143047, A143048, A143049, A213252, A213281, A213335.
Cf. A171200.

Programs

  • PARI
    a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^3);polcoeff(A,n)

Formula

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^3)^3.
G.f. satisfies: [A(x)^4 + A(-x)^4]/2 = [A(x)^3 + A(-x)^3]/2.
a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k>=0 and i+j+k=n-1} a(i) * a(j) * a(k). - Seiichi Manyama, Jul 08 2025

A143047 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^4.

Original entry on oeis.org

1, 1, -4, -10, 84, 265, -2604, -8900, 94692, 337940, -3767312, -13812674, 158785964, 593029550, -6967201736, -26372738120, 314904180100, 1204230041900, -14560722724912, -56130528427400, 685514219386576, 2659770565898729, -32749512944380172
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			A(x) = 1 + x - 4*x^2 - 10*x^3 + 84*x^4 + 265*x^5 - 2604*x^6 - 8900*x^7 +...
A(x)^4 = 1 + 4*x - 10*x^2 - 84*x^3 + 265*x^4 + 2604*x^5 - 8900*x^6 -...
A(x)^5 = 1 + 5*x - 10*x^2 - 120*x^3 + 265*x^4 + 3906*x^5 - 8900*x^6 -...
Note that a bisection of A^5 equals a bisection of A^4.

Links

Crossrefs

Cf. A143045, A143046, A143048, A143049, A213252, A213281, A213335.
Cf. A171202.

Programs

  • PARI
    a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^4);polcoeff(A,n)

Formula

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^4)^4.
G.f. satisfies: [A(x)^5 + A(-x)^5]/2 = [A(x)^4 + A(-x)^4]/2.
a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 08 2025

A143048 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^5.

Original entry on oeis.org

1, 1, -5, -15, 165, 630, -8151, -33780, 474045, 2052495, -30206330, -134392230, 2040588775, 9248893360, -143569282680, -659546365020, 10407737293965, 48303692377425, -771991701692175, -3611789245335285, 58311219888996170, 274581478640096340
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			A(x) = 1 + x - 5*x^2 - 15*x^3 + 165*x^4 + 630*x^5 - 8151*x^6 -++-...
A(x)^5 = 1 + 5*x - 15*x^2 - 165*x^3 + 630*x^4 + 8151*x^5 - 33780*x^6 -...
A(x)^6 = 1 + 6*x - 15*x^2 - 220*x^3 + 630*x^4 + 11286*x^5 - 33780*x^6 -...
Note that a bisection of A^6 equals a bisection of A^5.

Links

Crossrefs

Cf. A143045, A143046, A143047, A143049, A213252, A213281, A213335.
Cf. A171204.

Programs

  • PARI
    a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^5);polcoeff(A,n)

Formula

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^5)^5.
G.f. satisfies: [A(x)^6 + A(-x)^6]/2 = [A(x)^5 + A(-x)^5]/2.
a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025

A171206 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^6.

Original entry on oeis.org

1, 1, 12, 348, 19744, 2108784, 428817600, 169398274624, 131889504749568, 203937600707475456, 628561895904796999680, 3868208404121906515820544, 47571342639450113377565933568, 1169589733863427138021074362433536, 57499379103783344787572704263568097280, 5652994168279651703590653986228287051923456
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171200-A171205, A171207, A171208-A171211, A343439.
Cf. A143049, A171195.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1+x*A[2x]^6 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^6); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 2025

A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

Original entry on oeis.org

1, 1, 1, 7, 13, 112, 247, 2310, 5525, 53998, 135408, 1360289, 3518515, 36017352, 95223414, 988172368, 2655417765, 27844071255, 75769712590, 801012669457, 2201663313200, 23428926096576, 64924369564353, 694644371065372, 1938034271677595, 20829931845958872, 58448142042957576
Offset: 0

Views

Author

N. J. A. Sloane, Jul 12 2011

Keywords

Comments

The six sequences displayed in Table 1 of the Bousquet-Lamathe reference are A047749, A143546, A143547, A143554, A192893, A192894. From this one should be able to guess a g.f.

Links

Crossrefs

Column k=13 of A369929 and k=14 of A370062.
Cf. A143049.

Formula

From Seiichi Manyama, Jul 07 2025: (Start)
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x)*A(-x))^6 ).
G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^13.
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+2*(x_2+x_3+...+x_7)=n-1} a(x_1) * Product_{k=2..7} a(2*x_k). (End)
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_13>=0 and x_1+x_2+...+x_13=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5+x_6) * Product_{k=1..13} a(x_k). - Seiichi Manyama, Jul 09 2025

Extensions

a(11) onwards from Andrew Howroyd, Jan 26 2024
a(0)=1 prepended by Seiichi Manyama, Jul 07 2025
Showing 1-6 of 6 results.

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

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