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.

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.

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),", "))

A302355 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of the g.f. of triangular numbers (A000217).

Original entry on oeis.org

1, 6, 72, 1390, 37515, 1307691, 56000728, 2847503268, 167737660533, 11236731677941, 843757483026150, 70200772129462767, 6410711453857626149, 637516967943664853331, 68581800216461580653064, 7935677122691714769565104, 982824624566131043920711329, 129722104862557293606783635718
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Examples

			The initial coefficients of successive iterations of g.f. A(x) = x/(1 - x)^3 are as follows:
n = 1: 0, (1),  3,    6,    10,     15,  ... g.f. A(x)
n = 2: 0,  1,  (6),  30,   137,    588,  ... g.f. A(A(x))
n = 3: 0,  1,   9,  (72),  543,   3933,  ... g.f. A(A(A(x)))
n = 4: 0,  1,  12,  132, (1390), 14208,  ... g.f. A(A(A(A(x))))
n = 5: 0,  1,  15,  210,  2840, (37515), ... g.f. A(A(A(A(A(x)))))

Links

Crossrefs

Cf. A000217, A030280, A119821, A302356.

Programs

  • Mathematica
    Table[SeriesCoefficient[Nest[Function[x, x/(1 - x)^3], x, n], {x, 0, n}], {n, 18}]

A302356 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of the g.f. of squares (A000290).

Original entry on oeis.org

1, 8, 123, 3064, 107355, 4880896, 273564907, 18252720536, 1413701944431, 124714304306536, 12347969626724127, 1356049318451627812, 163596640499821625005, 21508738592360523314552, 3060986664449504902865167, 468816798653492762623354936, 76889830949170048691162162275
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Examples

			The initial coefficients of successive iterations of g.f. A(x) = x*(1 + x)/(1 - x)^3 are as follows:
n = 1: 0, (1),  4,    9,    16,      25,  ... g.f. A(x)
n = 2: 0,  1,  (8),  50,   276,    1397,  ... g.f. A(A(x))
n = 3: 0,  1,  12, (123), 1164,   10420,  ... g.f. A(A(A(x)))
n = 4: 0,  1,  16,  228, (3064),  39542,  ... g.f. A(A(A(A(x))))
n = 5: 0,  1,  20,  365,  6360, (107355), ... g.f. A(A(A(A(A(x)))))

Links

Crossrefs

Cf. A000290, A030279, A119821, A302355.

Programs

  • Mathematica
    Table[SeriesCoefficient[Nest[Function[x, x (1 + x)/(1 - x)^3], x, n], {x, 0, n}], {n, 17}]

A167014 Coefficients of x^n in the (n-1)-th iteration of x/(1-x)^2 for n>=1.

Original entry on oeis.org

1, 2, 14, 174, 3102, 72090, 2066073, 70420126, 2782399406, 125043232266, 6298621761335, 351484024822788, 21523422229105305, 1434943982960717580, 103463060233836392794, 8022156135358465396630, 665603454632822500855294
Offset: 0

Views

Author

Paul D. Hanna, Oct 28 2009

Keywords

Examples

			Let F_n(x) denote the n-th iteration of F(x) = x/(1-x)^2;
then coefficients in the successive iterations of F(x) begin:
F_0: [(1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
F(x):[1, (2), 3, 4, 5, 6, 7, 8, 9, 10, 11, ...];
F_2: [1, 4, (14), 46, 145, 444, 1331, 3926, 11434, 32960, ...];
F_3: [1, 6, 33, (174), 892, 4480, 22149, 108144, 522685, ...];
F_4: [1, 8, 60, 436, (3102), 21728, 150350, 1030158, 7000917, ...];
F_5: [1, 10, 95, 880, 8015, (72090), 642031, 5671962, ...];
F_6: [1, 12, 138, 1554, 17255, 189596, (2066073), 22361904, ...];
F_7: [1, 14, 189, 2506, 32830, 426244, 5494797, (70420126),...];
F_8: [1, 16, 248, 3784, 57132, 855840, 12740700, 188692708, (2782399406), ...]; ...
where the coefficients along the diagonal (shown in parenthesis)
form the initial terms of this sequence.

Crossrefs

Cf. A119821, A119820.

Programs

  • PARI
    {a(n)=local(F=x/(1-x+x*O(x^n))^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)))}

A302357 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of the g.f. of Fibonacci numbers (A000045).

Original entry on oeis.org

1, 2, 12, 102, 1165, 16603, 283283, 5624556, 127309302, 3234191224, 91094448874, 2816800580606, 94848640788603, 3454303753062123, 135278798460362984, 5668566821430630300, 253050028467629998389, 11988740253545762393562
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Examples

			The initial coefficients of successive iterations of g.f. A(x) = x/(1 - x - x^2) are as follows:
n = 1: 0, (1), 1,   2,    3,     5,  ... g.f. A(x)
n = 2: 0,  1, (2),  6,   17,    50,  ... g.f. A(A(x))
n = 3: 0,  1,  3, (12),  48,   197,  ... g.f. A(A(A(x)))
n = 4: 0,  1,  4,  20, (102),  532,  ... g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  30,  185, (1165), ... g.f. A(A(A(A(A(x)))))

Links

Crossrefs

Cf. A000045, A119821, A158832, A270863, A283679.

Programs

  • Mathematica
    Table[SeriesCoefficient[Nest[Function[x, x/(1 - x - x^2)], x, n], {x, 0, n}], {n, 18}]
Showing 1-6 of 6 results.

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