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 1-3 of 3 results.

A177395 G.f. satisfies: x = A(x) - A(A(x))^2 - A(A(A(x)))^3.

Original entry on oeis.org

1, 1, 5, 37, 338, 3530, 40546, 500781, 6556080, 90097535, 1290778689, 19180015667, 294460699563, 4656776745569, 75682133890995, 1261603117268148, 21537605020132685, 376060923637721700, 6708681746445946648
Offset: 1

Views

Author

Paul D. Hanna, May 29 2010

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 37*x^4 + 338*x^5 + 3530*x^6 +...
Coefficients in the iterations A_{n}(x), n=1..9, of A(x) begin:
A_1: [1, 1, 5, 37, 338, 3530, 40546, 500781, ...];
A_2: [1, 2, 12, 100, 998, 11197, 136682, 1780674, ...];
A_3: [1, 3, 21, 195, 2120, 25571, 332664, 4589974, ...];
A_4: [1, 4, 32, 328, 3868, 50078, 694340, 10157760, ...];
A_5: [1, 5, 45, 505, 6430, 89120, 1315126, 20388639, ...];
A_6: [1, 6, 60, 732, 10018, 148195, 2322702, 38106722, ...];
A_7: [1, 7, 77, 1015, 14868, 234017, 3886428, 67351872, ...];
A_8: [1, 8, 96, 1360, 21240, 354636, 6225480, 113733264, ...];
A_9: [1, 9, 117, 1773, 29418, 519558, 9617706, 184845297,...].
Coefficients in functions: x = A(x) - A_2(x)^2 - A_3(x)^3 begin:
(A_1)^1: [1, 1, 5, 37, 338, 3530, 40546, 500781, 6556080, ...];
(A_2)^2: [0, 1, 4, 28, 248, 2540, 28786, 352104, 4576404 ...];
(A_3)^3: [0, 0, 1,. 9,. 90,. 990, 11760, 148677, 1979676, ...].
Coefficients in functions: A(x) = A_2(x) - A_3(x)^2 - A_4(x)^3 begin:
(A_2)^1: [1, 2, 12, 100, 998, 11197, 136682, 1780674, 24453430, ...];
(A_3)^2: [0, 1,. 6,. 51, 516,. 5851,. 72052,. 945819, 13076714, ...];
(A_4)^3: [0, 0,. 1,. 12, 144,. 1816,. 24084,. 334074,. 4820636, ...].
Coefficients in functions: A_2(x) = A_3(x) - A_4(x)^2 -A_5(x)^3 begin:
(A_3)^1: [1, 3, 21, 195, 2120, 25571, 332664, 4589974, 66441348, ...];
(A_4)^2: [0, 1,. 8,. 80,. 912, 11384, 152092, 2144440, 31612640, ...];
(A_5)^3: [0, 0,. 1,. 15,. 210,. 2990,. 43890,. 664860, 10375278, ...].

Links

Crossrefs

Cf. A139702, A177396, A171780.

Programs

  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x-(A+x*O(x^n))^2-subst(A,x,A+x*O(x^n))^3)); polcoeff(A, n))}

Formula

G.f. satisfies: x = A( x - A(x)^2 - A(A(x))^3 ).
...
G.f. satisfies: A_{n}(x) = A_{n+1}(x) - A_{n+2}(x)^2 - A_{n+3}(x)^3 where A_{n+1}(x) = A_{n}(A(x)) denotes iteration with A_0(x)=x.
...
Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
. A = 1 + xB^2 + x^2*C^3;
. B = A + xC^2 + x^2*D^3;
. C = B + xD^2 + x^2*E^3;
. D = C + xE^2 + x^2*F^3; ...
. also B = A(A(x))/x, C = A(A(A(x)))/x, D = A(A(A(A(x))))/x, etc.

Extensions

Typos in examples corrected by Paul D. Hanna, May 29 2010
Formula corrected by Paul D. Hanna, May 29 2010

A177396 G.f. satisfies: x = A(x) - 2*A(A(x))^2 + A(A(A(x)))^3.

Original entry on oeis.org

1, 2, 15, 166, 2253, 34860, 592549, 10828938, 209868510, 4273308410, 90816552106, 2004641983826, 45791082809343, 1079342545547998, 26193557661047655, 653283080573467694, 16720129397788274155, 438610481782905122800
Offset: 1

Views

Author

Paul D. Hanna, May 29 2010

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 15*x^3 + 166*x^4 + 2253*x^5 + 34860*x^6 +...
Coefficients in the iterations A_{n}(x), n=1..9, of A(x) begin:
A_1: [1, 2, 15, 166, 2253, 34860, 592549, 10828938, ...];
A_2: [1, 4, 38, 490, 7473, 127274, 2349323, 46176042, ...];
A_3: [1, 6, 69, 1020, 17380, 325672, 6545871, 139035872, ...];
A_4: [1, 8, 108, 1804, 34078, 699716, 15287390, 350846310, ...];
A_5: [1, 10, 155, 2890, 60055, 1344140, 31807669, 786868272, ...];
A_6: [1, 12, 210, 4326, 98183, 2382590, 60814113, 1616326636, ...];
A_7: [1, 14, 273, 6160, 151718, 3971464, 108878847, 3097957506, ...];
A_8: [1, 16, 344, 8440, 224300, 6303752, 184875900, 5611606932, ...];
A_9: [1, 18, 423, 11214, 319953, 9612876, 300464469, 9696526206,...].
Coefficients in functions: x = A(x) - 2*A_2(x)^2 + A_3(x)^3 begin:
(A_1)^1: [1, 2, 15, 166, 2253, 34860, 592549, 10828938, ...];
(A_2)^2: [0, 1,. 8,. 92, 1284, 20310, 351572,. 6524886, ...];
(A_3)^3: [0, 0,. 1,. 18,. 315,. 5760, 110595,. 2220834, ...].
Coefficients in functions: A(x) = A_2(x) - 2*A_3(x)^2 +A_4(x)^3 begin:
(A_2)^1: [1, 4, 38, 490, 7473, 127274, 2349323, 46176042, ...];
(A_3)^2: [0, 1, 12, 174, 2868,. 51761, 1000664, 20438646, ...];
(A_4)^3: [0, 0,. 1,. 24,. 516,. 11108,. 244554,. 5530188, ...].
Coefficients in functions: A_2(x) = A_3(x) -2*A_4(x)^2 +A_5(x)^3 begin:
(A_3)^1: [1, 6, 69, 1020, 17380, 325672, 6545871, 139035872, ...];
(A_4)^2: [0, 1, 16,. 280,. 5336, 108684, 2334344,. 52385500, ...];
(A_5)^3: [0, 0,. 1,.. 30,.. 765,. 18970,. 472140,. 11911170, ...].

Links

Crossrefs

Cf. A139702, A177395, A171780.

Programs

  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - 2*(A+x*O(x^n))^2+subst(A,x,A+x*O(x^n))^3)); polcoeff(A, n))}

Formula

G.f. satisfies: x = A( x - 2*A(x)^2 + A(A(x))^3 ).
...
G.f. satisfies: A_{n}(x) = A_{n+1}(x) - 2*A_{n+2}(x)^2 + A_{n+3}(x)^3 where A_{n+1}(x) = A_{n}(A(x)) denotes iteration with A_0(x)=x.
...
Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
. A = 1 + 2xB^2 - x^2*C^3;
. B = A + 2xC^2 - x^2*D^3;
. C = B + 2xD^2 - x^2*E^3;
. D = C + 2xE^2 - x^2*F^3; ...
also B = A(A(x))/x, C = A(A(A(x)))/x, D = A(A(A(A(x))))/x, etc.

Extensions

Formula corrected by Paul D. Hanna, May 29 2010

A172387 G.f. satisfies: A(x) = x + x*Sum_{n>=1} G_{n}(x)^n, where G_{n}(x) is the n-th iteration of A(x) defined by G{n}(x) = G_{n-1}(A(x)) with G_0(x)=x.

Original entry on oeis.org

1, 1, 2, 7, 33, 187, 1208, 8626, 66782, 553355, 4862938, 45025668, 437029462, 4429568600, 46738108168, 512097036882, 5814415004953, 68293044651990, 828547526906649, 10369747261970151, 133732024450930096
Offset: 0

Views

Author

Paul D. Hanna, Feb 03 2010

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 33*x^5 + 187*x^6 +...
Related expansions:
A(A(x)) = x + 2*x^2 + 6*x^3 + 25*x^4 + 130*x^5 + 789*x^6 +...
A(A(x))^2 = x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 396*x^6 +...
A(A(A(x))) = x + 3*x^2 + 12*x^3 + 60*x^4 + 353*x^5 + 2348*x^6 +...
A(A(A(x)))^3 = x^3 + 9*x^4 + 63*x^5 + 423*x^6 + 2895*x^7 +...
Coefficients in the n-th iteration of the g.f. begin:
n=1: [1, 1, 2, 7, 33, 187, 1208, 8626, 66782, 553355, ...];
n=2: [1, 2, 6, 25, 130, 789, 5376, 40140, 323198, 2775204, ...];
n=3: [1, 3, 12, 60, 353, 2348, 17208, 136574, 1159754, ...];
n=4: [1, 4, 20, 118, 788, 5800, 46236, 394382, 3568108, ...];
n=5: [1, 5, 30, 205, 1545, 12595, 109664, 1010914, 9803334, ...];
n=6: [1, 6, 42, 327, 2758, 24817, 235932, 2354038, 24532158, ...];
n=7: [1, 7, 56, 490, 4585, 45304, 469000, 5059950, 56677550, ...];
n=8: [1, 8, 72, 700, 7208, 77768, 873352, 10164212, 122210376, ...];
n=9: [1, 9, 90, 963, 10833, 126915, 1539720, 19271058, 248179134, ...];
n=10:[1, 10, 110, 1285, 15690, 198565, 2591528, 34766008, 478309118, ...]; ...
Coefficients in the n-th power of the n-th iteration of the g.f. begin:
n=1: [1, 1, 2, 7, 33, 187, 1208, 8626, 66782, 553355, ...];
n=2: [0, 1, 4, 16, 74, 396, 2398, 16093, 117752, 927818, ...];
n=3: [0, 0, 1, 9, 63, 423, 2895, 20634, 154323, 1213566, ...];
n=4: [0, 0, 0, 1, 16, 176, 1688, 15312, 136320, 1214472, ...];
n=5: [0, 0, 0, 0, 1, 25, 400, 5275, 62850, 707350, 7710070, ...];
n=6: [0, 0, 0, 0, 0, 1, 36, 792, 13842, 212028, 2989698, ...];
n=7: [0, 0, 0, 0, 0, 0, 1, 49, 1421, 31899, 614166, 10685675, ...];
n=8: [0, 0, 0, 0, 0, 0, 0, 1, 64, 2368, 66528, 1577280, ...];
n=9: [0, 0, 0, 0, 0, 0, 0, 0, 1, 81, 3726, 128223, 3676887, ...];
n=10:[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 100, 5600, 231850, 7917900, ...]; ...
Column sums in the above table form this sequence shift left 1 place.

Crossrefs

Cf. A171780 (variant).

Programs

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

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