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 1-2 of 2 results.

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

A177397 G.f. satisfies: x = A(x) - A(A(x))^2 - A(A(A(x)))^2.

Original entry on oeis.org

1, 2, 20, 316, 6312, 146256, 3765792, 105104272, 3130299744, 98434722240, 3243746014592, 111400312737152, 3970597596057856, 146403897677390336, 5570169496704513024, 218228733514994839808, 8789314898568643716608
Offset: 1

Views

Author

Paul D. Hanna, May 31 2010

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 20*x^3 + 316*x^4 + 6312*x^5 + 146256*x^6 +...
Related expansions:
A(A(x)) = x + 4*x^2 + 48*x^3 + 840*x^4 + 18016*x^5 + 440992*x^6 +...
A(A(A(x))) = x + 6*x^2 + 84*x^3 + 1620*x^4 + 37352*x^5 +969328*x^6 +...
A_{-1}(x) = x - 2*x^2 - 12*x^3 - 156*x^4 - 2776*x^5 - 59344*x^6 -...
A_{-2}(x) = x - 4*x^2 - 16*x^3 - 200*x^4 - 3488*x^5 - 73632*x^6 -...
...
Illustrate A_{n}(x) = A_{n+1}(x) - A_{n+2}(x)^2 - A_{n+3}(x)^2 by the following tables of coefficients in the iterations of g.f. A(x).
Coefficients in iterations A_{n}(x), for n=1..8, begin:
A_1: [1, 2, 20, 316, 6312, 146256, 3765792, 105104272,...];
A_2: [1, 4, 48, 840, 18016, 440992, 11875712, 344335328,...];
A_3: [1, 6, 84, 1620, 37352, 969328, 27429152, 830501936,...];
A_4: [1, 8, 128, 2704, 66944, 1843776, 54945792, 1742374336,...];
A_5: [1, 10, 180, 4140, 109800, 3208080, 100748064, 3350443472,...];
A_6: [1, 12, 240, 5976, 169312, 5241056, 173389696, 6048725920,...];
A_7: [1, 14, 308, 8260, 249256, 8160432, 284130336, 10393259632,...];
A_8: [1, 16, 384, 11040, 353792, 12226688, 447456256, 17147935616,...].
...
Coefficients in squared iterations A_{n}(x)^2, for n=1..8, begin:
(A_1)^2: [0, 1, 4, 44, 712, 14288, 330400, 8468944, 235111136,...];
(A_2)^2: [0, 1, 8, 112, 2064, 45056, 1106752, 29714496, 856278464,...];
(A_3)^2: [0, 1, 12, 204, 4248, 101200, 2659040, 75389776, ...];
(A_4)^2: [0, 1, 16, 320, 7456, 193536, 5450880, 163841280, ...];
(A_5)^2: [0, 1, 20, 460, 11880, 334800, 10102560, 322325328, ...];
(A_6)^2: [0, 1, 24, 624, 17712, 539648, 17414080, 589547072, ...];
(A_7)^2: [0, 1, 28, 812, 25144, 824656, 28388192, 1018522064, ...];
(A_8)^2: [0, 1, 32, 1024, 34368, 1208320, 44253440, 1679760384, ...].

Crossrefs

Cf. A177395, A177396.

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))^2)); polcoeff(A, n))}

Formula

G.f. A(x) satisfies: A_{n}(x) = A_{n+1}(x) - A_{n+2}(x)^2 - A_{n+3}(x)^2 where A_{n+1}(x) = A_{n}(A(x)) denotes iteration with A_0(x)=x.
G.f. satisfies: A(x) = A(A(x)) - A(A(A(x)))^2 - A(A(A(A(x))))^2.
G.f. satisfies: x = A( x - A(x)^2 - A(A(x))^2 ).
...
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 + xC^2;
. B = A + xC^2 + xD^2;
. C = B + xD^2 + xE^2;
. D = C + xE^2 + xF^2; ...
. also B = A(A(x))/x, C = A(A(A(x)))/x, D = A(A(A(A(x))))/x, etc.
Showing 1-2 of 2 results.

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