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.

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A112320 Coefficient of x^n in the (n+1)-th iteration of (x + x^2) for n>=1.

Original entry on oeis.org

1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, 10052947476, 261595087182, 7509722346204, 235808741944100, 8040824716606176, 295914258931377276, 11690732617035570008, 493527339623630078552
Offset: 1

Views

Author

Paul D. Hanna, Sep 06 2005

Keywords

Examples

			The first few iterations of (x+x^2) begin:
F(x) = x + x^2;
F(F(x)) = (1)*x + 2*x^2 + 2*x^3 + x^4;
F(F(F(x))) = x + (3)*x^2 + 6*x^3 + 9*x^4 + 10*x^5 +...;
F(F(F(F(x)))) = x + 4*x^2 + (12)*x^3 + 30*x^4 + 64*x^5 +...;
F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + (70)*x^4 + 220*x^5 +...;
F(F(F(F(F(F(x)))))) = x + 6*x^2 + 30*x^3 + 135*x^4 + (560)*x^5 +...;
coefficients enclosed in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A112317, A112319.

Programs

  • PARI
    {a(n)=local(F=x+x^2, G=x+x*O(x^n));if(n<1,0, for(i=1,n+1,G=subst(F,x,G));return(polcoeff(G,n,x)))}
for(n=1,25,print1(a(n),", "))

Formula

a(n) = [x^n] F_{n+1}(x) where F_{n+1}(x) = F_n(x+x^2) with F_1(x) = x+x^2 and F_0(x)=x for n>=1.

A119818 a(n) is the least nonnegative integer that can appear as the coefficient of x^n in the n-th iteration of any integer function that begins with the same initial n-1 terms as the g.f. of A119817 for n>1, with a(1)=1.

Original entry on oeis.org

1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 7, 12, 0, 6, 0, 9, 2, 11, 0, 8, 10, 13, 18, 18, 0, 0, 0, 0, 1, 0, 4, 6, 0, 19, 22, 1, 0, 41, 0, 14, 4, 23, 0, 26, 21, 22, 14, 11, 0, 42, 10, 21, 38, 0, 0, 46, 0, 31, 9, 40, 8, 33, 0, 16, 35, 7, 0, 66, 0, 37, 20, 63, 20, 58, 0, 74, 9, 0, 0, 23, 5, 0, 31, 75
Offset: 1

Views

Author

Paul D. Hanna, May 31 2006

Keywords

Comments

For prime p, a(p) = 0; for all n>=1, 0 <= a(n) < n.

Examples

			Let F(x) = g.f. of A119817 = [1,1,-2,8,-40,210,-1032,4074,-9084,...],
then the coefficient of x^n in the n-th iteration of F(x)
forms [1,2,0,2,0,0,0,0,0,0,0,10,...], as illustrated by:
F(x) = (1)x + x^2 - 2x^3 + 8x^4 - 40x^5 + 210x^6 - 1032x^7 + 4074x^8+..
F(F(x)) = x + (2)x^2 - 2x^3 + 7x^4 - 30x^5 + 118x^6 -268x^7 -1430x^8+..
F(F(F(x))) = x + 3x^2 + (0)x^3 + 3x^4 -12x^5 +18x^6 +240x^7 -3119x^8+..
F(F(F(F(x)))) = x + 4x^2 + 4x^3 + (2)x^4 - 4x^5 - 18x^6 + 276x^7+...
F(F(F(F(F(x))))) = x + 5x^2 + 10x^3 + 10x^4 +(0)x^5 -20*x^6 +128*x^7+..
F(F(F(F(F(F(x)))))) = x + 6x^2 + 18x^3 +33x^4 +30x^5 +(0)x^6 -24x^7+..

Crossrefs

Cf. A119817, A119816, A112317.

Programs

  • PARI
    {a(n)=my(A=vector(n),B,F=x+x^2,G); if(n==1||n==2,n,A[1]=1; A[2]=1; B=A; B[2]=2; for(m=3,n,G=x+x*O(x^n); for(k=1,m,G=subst(F,x,G)); B[m]=polcoeff(G,m,x); A[m]=(m-1-B[m])\m; F=F+A[m]*x^m); return(B[n]+n*A[n]))}

Formula

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F(x)) such that F(x) = g.f. of A119817 causes {a(n)} to be the least nonnegative integers.

A194971 a(n) equals the coefficient of x^(2*n-1) in the n-th iteration of x+x^2 for n>=1.

Original entry on oeis.org

1, 2, 10, 188, 8994, 832680, 127104492, 28951041456, 9201410927608, 3889680139527920, 2109876998624179100, 1428197506614652750656, 1179911974067256647171268, 1168294604146384807206421176, 1365624160842343461171218423880
Offset: 1

Views

Author

Paul D. Hanna, Sep 06 2011

Keywords

Examples

			The coefficients of x^k, k>=1, in the n-th iterations of x+x^2 begin:
n=1: [(1), 1];
n=2: [1, 2, (2), 1];
n=3: [1, 3, 6, 9, (10), 8, 4, 1];
n=4: [1, 4, 12, 30, 64, 118, (188), 258, 302, 298, 244,162,84,32,8,1];
n=5: [1, 5, 20, 70, 220, 630, 1656, 4014, (8994), 18654, 35832,...];
n=6: [1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, (832680), ...]; ...
coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A194972, A122888, A112317.

Programs

  • PARI
    {a(n)=local(A=x,G=x+x^2); for(i=1,n, A=subst(G, x, A+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}

A194972 a(n) equals the coefficient of x^n in the (2*n-1)-th iteration of x+x^2 for n>=1.

Original entry on oeis.org

1, 3, 20, 231, 3864, 85140, 2332616, 76485227, 2921536088, 127421864328, 6248486040840, 340321635330534, 20383240346962440, 1331538898625750100, 94216429100347571448, 7178425650032302557691, 585936966156456139931584, 51011156414845408925712816
Offset: 1

Views

Author

Paul D. Hanna, Sep 06 2011

Keywords

Examples

			The coefficients of x^k, k>=1, in the odd iterations of x+x^2 begin:
n=1: [(1), 1];
n=3: [1,(3), 6, 9, 10, 8, 4, 1];
n=5: [1, 5,(20), 70, 220, 630, 1656, 4014, 8994, 18654, ...];
n=7: [1, 7, 42,(231), 1190, 5810, 27076, 121023, 520626, ...];
n=9: [1, 9, 72, 540,(3864), 26628, 177744, 1153740, 7303164, ...];
n=11:[1, 11, 110, 1045, 9570,(85140), 739332, 6286797, ...];
n=13:[1, 13, 156, 1794, 20020, 218218,(2332616), 24519066, ...];
n=15:[1, 15, 210, 2835, 37310, 481390, 6110468,(76485227), ...]; ...
coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A194971, A122888, A112317.

Programs

  • PARI
    {a(n)=local(A=x,G=x+x^2); for(i=1,2*n-1, A=subst(G, x, A+x*O(x^n))); polcoeff(A,n)}
Previous Showing 11-14 of 14 results.

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