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.

A119816 Least positive integers that can appear as the coefficient of x^n the n-th iteration of an integer function F(x) where F(0)=0, for n>=1; the F(x) that satisfies this minimal condition is the g.f. of A119815.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 3, 9, 11, 4, 13, 11, 14, 8, 17, 4, 19, 4, 1, 4, 23, 24, 5, 17, 27, 22, 29, 16, 31, 32, 24, 4, 30, 36, 37, 4, 36, 10, 41, 18, 43, 41, 17, 27, 47, 40, 28, 29, 7, 10, 53, 9, 1, 24, 49, 4, 59, 57, 61, 35, 31, 48, 39, 16, 67, 24, 51, 9, 71, 46, 73, 4, 56, 11, 55, 62, 79
Offset: 1

Views

Author

Paul D. Hanna, May 31 2006

Keywords

Comments

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

Examples

			Let F(x) = g.f. of A119815 = [1,1,-1,1,1,-11,23,-20,731,-4860,...],
then the coefficient of x^n in the n-th iteration of F(x)
forms [1,2,3,4,5,4,7,8,3,9,11,...], as illustrated by:
F(x) = (1)x + x^2 - x^3 + x^4 + x^5 - 11x^6 + 23x^7 - 20x^8 + 731x^9+..
F(F(x)) = x + (2)x^2 - 2x^4 + 6x^5 - 8x^6 - 50x^7 + 78x^8 + 1688x^9+...
F(F(F(x))) = x + 3x^2 + (3)x^3 - 3x^4 - x^5 + 17x^6 - 81x^7 -370x^8+...
F(F(F(F(x)))) = x + 4x^2 + 8x^3 + (4)x^4 - 12x^5 + 4x^6 + 12x^7 +...
F(F(F(F(F(x))))) = x + 5x^2 + 15x^3 + 25x^4 + (5)x^5 - 55x^6 -33x^7+...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 24x^3 + 66x^4 + 106x^5 + (4)x^6 +...

Links

Crossrefs

Cf. A119815, A119818.

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-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 A119815 causes {a(n)} to be the least positive integers.

A119820 Coefficients of x^n in the n-th iteration of x*(1+x)^2 for n>=1.

Original entry on oeis.org

1, 4, 27, 300, 4790, 101010, 2660028, 84191772, 3115739358, 132074618544, 6311492388432, 335744715016854, 19678501474466211, 1260060524755139120, 87519840721085385096, 6553840567691077634748, 526360263009035464610574
Offset: 1

Views

Author

Paul D. Hanna, Jun 01 2006

Keywords

Examples

			The successive iterations of F(x) = x*(1+x)^2 begin:
F(x) = (1)x + 2x^2 + x^3
F(F(x)) = x + (4)x^2 + 10x^3 + 18x^4 + 23x^5 + 22x^6 + 15x^7 + 6x^8 +...
F(F(F(x))) = x + 6x^2 + (27)x^3 + 102x^4 + 333x^5 + 960x^6 + 2472x^7 +...
F(F(F(F(x)))) = x + 8x^2 + 52x^3 + (300)x^4 + 1578x^5 + 7692x^6 +...
F(F(F(F(F(x))))) = x + 10x^2 + 85x^3 + 660x^4 + (4790)x^5 + 32920x^6+...
F(F(F(F(F(F(x)))))) = x + 12x^2 +126x^3 +1230x^4+11385x^5+(101010)x^6+...

Crossrefs

Cf. A119821, A112317, A119819, A119817, A119815.

Programs

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

Formula

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F_1(x)) with F_1(x) = x*(1+x)^2.

A119821 Coefficients of x^n in the n-th iteration of x/(1-x)^2 for n>=1.

Original entry on oeis.org

1, 4, 33, 436, 8015, 189596, 5494797, 188692708, 7494744807, 338103170428, 17079035749061, 955117390512858, 58584586487137113, 3910851585418994256, 282272352712037938081, 21904366942822876046020
Offset: 1

Views

Author

Paul D. Hanna, Jun 01 2006

Keywords

Comments

The coefficient of x^n in the n-th iteration of x/(1-x) = n^(n-1) = A000169(n); does this variant have a simple formula for a(n)?

Examples

			The successive iterations of F(x) = x/(1-x)^2 begin:
F(x) = (1)x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 6x^6 + 7x^7 + 8x^8 +...
F(F(x)) = x + (4)x^2 + 14x^3 + 46x^4 + 145x^5 + 444x^6 + 1331x^7 +...
F(F(F(x))) = x + 6x^2 + (33)x^3 + 174x^4 + 892x^5 + 4480x^6 +...
F(F(F(F(x)))) = x + 8x^2 + 60x^3 + (436)x^4 + 3102x^5 + 21728x^6 +...
F(F(F(F(F(x))))) = x + 10x^2 + 95x^3 + 880x^4 + (8015)x^5 +72090x^6+..
F(F(F(F(F(F(x)))))) = x + 12x^2+138x^3+1554x^4+17255x^5+(189596)x^6+..

Crossrefs

Cf. A119820, A112317; A119819, A119817, A119815.

Programs

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

Formula

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F(x)) with F(x) = x/(1-x)^2.

A119817 Integer a(n) produces the least nonnegative integer coefficient of x^n in the n-th iteration of g.f. A(x).

Original entry on oeis.org

1, 1, -2, 8, -40, 210, -1032, 4074, -9084, -1485, -139344, -1178057, 97107644, 533077818, -43465435335, -997494915376, 35039558716800, 1885975569825115, -36684866143759995, -4946226556607087316, 24828007395162323458, 18213320246807011794109
Offset: 1

Views

Author

Paul D. Hanna, May 31 2006

Keywords

Examples

			The successive iterations of g.f. A(x) begin:
A(x) = (1)x + x^2 - 2x^3 + 8x^4 - 40x^5 + 210x^6 - 1032x^7 + 4074x^8+..
A(A(x)) = x + (2)x^2 - 2x^3 + 7x^4 - 30x^5 + 118x^6 -268x^7 -1430x^8+..
A(A(A(x))) = x + 3x^2 + (0)x^3 + 3x^4 -12x^5 +18x^6 +240x^7 -3119x^8+..
A(A(A(A(x)))) = x + 4x^2 + 4x^3 + (2)x^4 - 4x^5 - 18x^6 + 276x^7+...
A(A(A(A(A(x))))) = x + 5x^2 + 10x^3 + 10x^4 +(0)x^5 -20*x^6 +128*x^7+..
A(A(A(A(A(A(x)))))) = x + 6x^2 + 18x^3 +33x^4 +30x^5 +(0)x^6 -24x^7+..
Coefficients [x^n] of n-th iteration of A(x) forms A119818:
[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,...].

Crossrefs

Cf. A119818, A119815, A112317.

Programs

  • PARI
    {a(n)=local(F=x+x^2+sum(k=3,n-1,a(k)*x^k),G=x+x*O(x^n)); if(n<1,0,if(n<=2,1, for(k=1,n,G=subst(F,x,G)); return((n-1-polcoeff(G,n,x)) )))}

A119819 a(n) equals the coefficient of x^(n-1) in the (n-1)-th iteration of g.f. A(x) for n>1, with a(1)=1.

Original entry on oeis.org

1, 1, 2, 12, 138, 2370, 54190, 1553258, 53883088, 2211883428, 105760271082, 5819880201432, 364979361177134, 25865387272507770, 2056021496464455000, 182094050389241652004, 17861355920109599058260, 1929874166854161381238676, 228564755268775651632722308, 29540844190975459101114949972
Offset: 1

Views

Author

Paul D. Hanna, May 31 2006

Keywords

Comments

Here the zeroth iteration of A(x) equals x, the first iteration is itself, the 2nd iteration of A(x) = A(A(x)), etc.

Examples

			The coefficients in the n-th iteration of g.f. A(x) begin:
n=1: [1, 1,  2,   12,   138,   2370,   54190,  1553258,   53883088, ...];
n=2: [1, 2,  6,   35,   370,   6000,  132344,  3704032,  126318024, ...];
n=3: [1, 3, 12,   75,   758,  11612,  245746,  6688885,  223699238, ...];
n=4: [1, 4, 20,  138,  1388,  20322,  411708, 10854152,  354952262, ...];
n=5: [1, 5, 30,  230,  2370,  33760,  656414, 16711414,  532707614, ...];
n=6: [1, 6, 42,  357,  3838,  54190, 1018484, 25016120,  775036254, ...];
n=7: [1, 7, 56,  525,  5950,  84630, 1553258, 36874397, 1107956996, ...];
n=8: [1, 8, 72,  740,  8888, 128972, 2337800, 53883088, 1568966580, ...];
n=9: [1, 9, 90, 1008, 12858, 192102, 3476622, 78308058, 2211883428, ...]; ...
where the diagonal of coefficients equals this sequence shift left 1 place.
...
More explicitly, the successive iterations of g.f. A(x) begin:
A(x) = (1)x + x^2 + 2x^3 + 12x^4 + 138x^5 + 2370x^6 + 54190x^7 +...
A(A(x)) = x + (2)x^2 + 6x^3 + 35x^4 + 370x^5 + 6000x^6 + 132344x^7 +...
A(A(A(x))) = x + 3x^2 + (12)x^3 + 75x^4 + 758x^5 + 11612x^6 +...
A(A(A(A(x)))) = x + 4x^2 + 20x^3 + (138)x^4 + 1388x^5 + 20322x^6 +...
A(A(A(A(A(x))))) = x + 5x^2 + 30x^3 + 230x^4 + (2370)x^5 + 33760x^6+...
A(A(A(A(A(A(x)))))) = x + 6x^2 +42x^3 +357x^4 +3838x^5 + (54190)x^6+...
...

Crossrefs

Cf. A112317, A119820, A119821, A119815, A119817.

Programs

  • PARI
    {a(n)=local(F=x+x^2+sum(m=3,n-1,a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0,if(n<=2,1,for(i=1, n-1, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}
for(n=1,30,print1(a(n),", "))
Showing 1-5 of 5 results.

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