A112104 -id:A112104 - cp's OEIS frontend

A112127 G.f. A(x) satisfies A(A(A(..(A(x))..))) = B(x) (13th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,..,13}, with B(0) = 0.

1, 1, -11, 193, -4043, 92233, -2188907, 52544305, -1250612651, 29060631481, -651497950667, 13997025548641, -289196932607819, 5873067677083177, -122109541297984368, 2669034419647762122, -58550172867811577842, 1127335101086707607932

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 - 11*x^3 + 193*x^4 - 4043*x^5 + 92233*x^6 +...
where A(A(A(A(A(A(A(A(A(A(A(A(A(x))))))))))))) =
x + 13*x^2 + 13*x^3 + 13*x^4 + 13*x^5 + 13*x^6 + 13*x^7 +...
is the g.f. of A112126.

Cf. A112126, A112104-A112125.

{a(n,m=13)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); return(polcoeff(F,n,x)))}

A112105 G.f. A(x) satisfies A(A(x)) = B(x) such that the coefficients of B(x) consist of all 1's and 2's, with A(0) = 0.

Original entry on oeis.org

1, 1, 0, 0, 1, -3, 7, -10, -5, 84, -248, 90, 2160, -7541, -5846, 122824, -186259, -2036532, 8665409, 36714511, -345711246, -517802065, 14415153844, -9423161197, -653074772917, 1896978939457, 32374651932128, -184814895196023, -1733326272860598, 16839263882542991, 96742403684106435

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 + x^5 - 3*x^6 + 7*x^7 - 10*x^8 - 5*x^9 +...
where A(A(x)) = x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 +... is the g.f. of A112104.

Kenny Lau, Table of n, a(n) for n = 1..553

Cf. A112104, A112106-A112127.

kmax = 40;
A[x_] := Sum[a[k] x^k, {k, kmax}];
B[x_] := Sum[b[k] x^k, {k, kmax}];
sol = {a[1] -> 1, b[1] -> 1};
Do[sc = SeriesCoefficient[A[(A[x] /. sol) + O[x]^(k+1)] - B[x], {x, 0, k}] /. sol; r = Reduce[(b[k] == 1 || b[k] == 2) && sc == 0, {a[k], b[k]}, Integers]; sol = Join[r // ToRules, sol], {k, 2, kmax}];
a /@ Range[kmax] /. sol (* Jean-François Alcover, Nov 05 2019 *)

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

A112117 G.f. A(x) satisfies A(A(A(..(A(x))..))) = B(x) (8th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,..,8}, with B(0) = 0.

Original entry on oeis.org

1, 1, -6, 60, -720, 9398, -126958, 1719439, -22778647, 288721672, -3426131120, 37291873546, -368633930696, 3421668183648, -33763691015949, 382711017377914, -3403489111329505, -22613095886515578, 1672401759052466166, -27936127591842262118, -15637150116164531317

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 - 6*x^3 + 60*x^4 - 720*x^5 + 9398*x^6 +...
where A(A(A(A(A(A(A(A(x)))))))) =
x + 8*x^2 + 8*x^3 + 4*x^4 + 8*x^5 + 4*x^6 + 8*x^7 +...
is the g.f. of A112116.

Cf. A112116, A112104-A112115, A112118-A112127.

{a(n,m=8)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); return(polcoeff(F,n,x)))}

A112106 Unique sequence of numbers {1,2,3} where g.f. A(x) satisfies A(x) = B(B(B(x))) (3rd self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

Original entry on oeis.org

1, 3, 3, 3, 2, 2, 1, 2, 1, 3, 1, 1, 3, 3, 3, 2, 3, 3, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 3, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 1, 3, 2, 1, 3, 2, 2, 1, 2, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 2, 3, 3, 3, 3, 3, 3, 1, 1, 2, 2, 3, 3, 1, 3, 2, 1, 2, 2, 1, 1, 3, 1

Offset: 1

Paul D. Hanna, Aug 27 2005

nonn

			G.f.: A(x) = x + 3*x^2 + 3*x^3 + 3*x^4 + 2*x^5 + 2*x^6 + ...
then A(x) = B(B(B(x))) where
B(x) = x + x^2 - x^3 + 3*x^4 - 10*x^5 + 35*x^6 - 119*x^7 + ...
is the g.f. of A112107.

Cf. A112107, A112104, A112105, A112108-A112127.

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

A112107 G.f. A(x) satisfies A(A(A(x))) = B(x) (3rd self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3}, with B(0) = 0.

Original entry on oeis.org

1, 1, -1, 3, -10, 35, -119, 360, -792, -33, 12779, -82525, 305861, -552011, -126321, -8385020, 138177591, -433073834, -5366414982, 51203452090, 123835509276, -4174647911014, 5274854624423, 373574363131841, -2054088594386738, -34047892948849106, 391005463740951942

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 - x^3 + 3*x^4 - 10*x^5 + 35*x^6 - 119*x^7 + ...
where A(A(A(x))) = x + 3*x^2 + 3*x^3 + 3*x^4 + 2*x^5 + ...
is the g.f. of A112106.

Cf. A112106, A112104, A112105, A112108-A112127.

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

A112108 Unique sequence of numbers {1,2,3,4} where g.f. A(x) satisfies A(x) = B(B(B(B(x)))) (4th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

Original entry on oeis.org

1, 4, 4, 2, 4, 2, 4, 4, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 3, 4, 3, 2, 4, 1, 2, 4, 2, 3, 1, 4, 2, 4, 3, 1, 4, 4, 4, 2, 2, 2, 3, 3, 2, 3, 2, 2, 4, 1, 4, 2, 2, 1, 4, 3, 3, 3, 1, 1, 3, 3, 4, 4, 3, 3, 3, 3, 1, 4, 4, 3, 2, 4, 2, 2, 2, 1, 3, 4, 2, 3, 3, 1, 4, 2, 3, 1, 1, 3, 3, 4, 2, 4, 3, 1, 4, 3, 2, 1, 1, 1, 2, 1, 4, 4

Offset: 1

Paul D. Hanna, Aug 27 2005

nonn

			G.f.: A(x) = x + 4*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + ...
then A(x) = B(B(B(B(x)))) where
B(x) = x + x^2 - 2*x^3 + 8*x^4 - 38*x^5 + 194*x^6 - 992*x^7 + ...
is the g.f. of A112109.

Cf. A112109, A112104-A112107, A112110-A112127.

{a(n,m=4)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}

A112109 G.f. A(x) satisfies A(A(A(A(x)))) = B(x) (4th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,4}, with B(0) = 0.

Original entry on oeis.org

1, 1, -2, 8, -38, 194, -992, 4777, -19831, 56116, 48008, -2062286, 16053636, -70193968, 155216743, -968038798, 23817048561, -233579083166, 333773365, 21684104628935, -121906541882294, -2171063003748135, 30425707365005935, 192144123118329872

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 - 2*x^3 + 8*x^4 - 38*x^5 + 194*x^6 - 992*x^7 +...
where A(A(A(A(x)))) =
x + 4*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 4*x^8 +...
is the g.f. of A112108.

Cf. A112108, A112104-A112107, A112110-A112127.

{a(n,m=4)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); return(polcoeff(F,n,x)))}

A112110 Unique sequence of numbers {1,2,3,4,5} where g.f. A(x) satisfies A(x) = B(B(B(B(B(x))))) (5th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

Original entry on oeis.org

1, 5, 5, 5, 5, 5, 4, 4, 4, 4, 3, 1, 1, 1, 5, 3, 1, 1, 5, 3, 4, 3, 2, 1, 5, 4, 1, 4, 1, 5, 1, 4, 5, 4, 2, 1, 5, 2, 5, 4, 5, 5, 4, 1, 1, 5, 4, 3, 5, 1, 5, 2, 2, 3, 1, 3, 2, 5, 2, 5, 3, 2, 3, 5, 2, 1, 2, 3, 1, 5, 1, 4, 5, 4, 3, 3, 2, 4, 2, 3, 4, 5, 2, 5, 5, 2, 4, 2, 3, 5, 3, 2, 4, 2, 2, 1, 1, 2, 3, 4, 5, 3, 3, 1, 5

Offset: 1

Paul D. Hanna, Aug 27 2005

nonn

			G.f.: A(x) = x + 5*x^2 + 5*x^3 + 5*x^4 + 5*x^5 + 5*x^6 + ...
then A(x) = B(B(B(B(B(x))))) where
B(x) = x + x^2 - 3*x^3 + 17*x^4 - 115*x^5 + 841*x^6 + ...
is the g.f. of A112111.

Cf. A112111, A112104-A112109, A112112-A112127.

{a(n,m=5)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}

A112111 G.f. A(x) satisfies A(A(A(A(A(x))))) = B(x) (5th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,4,5}, with B(0) = 0.

Original entry on oeis.org

1, 1, -3, 17, -115, 841, -6288, 46174, -320366, 1997348, -10216611, 32418767, 68603755, -1909624513, 15239954041, -103620859984, 1499179409198, -25808959095992, 258001631302410, -239530586418995, -25424691109062239, 84868851253494310

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 - 3*x^3 + 17*x^4 - 115*x^5 + 841*x^6 -6288*x^7 +...
where A(A(A(A(A(x))))) =
x + 5*x^2 + 5*x^3 + 5*x^4 + 5*x^5 + 5*x^6 + 4*x^7 + 3*x^8 +...
is the g.f. of A112110.

Cf. A112110, A112104-A112109, A112112-A112127.

{a(n,m=5)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); return(polcoeff(F,n,x)))}

A112112 Unique sequence of numbers {1,2,3,...,6} where g.f. A(x) satisfies A(x) = B(B(B(..(B(x))..))) (6th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

Original entry on oeis.org

1, 6, 6, 3, 4, 4, 6, 2, 5, 3, 3, 5, 3, 2, 5, 3, 3, 4, 5, 4, 3, 2, 6, 4, 3, 6, 2, 5, 6, 4, 2, 5, 4, 5, 1, 1, 1, 4, 4, 2, 3, 6, 6, 5, 5, 4, 3, 5, 5, 2, 2, 1, 3, 6, 1, 5, 2, 6, 5, 4, 3, 4, 6, 6, 5, 5, 6, 1, 5, 6, 6, 3, 3, 1, 5, 4, 5, 1, 5, 2, 2, 4, 3, 4, 2, 1, 6, 1, 3, 2, 4, 1, 3, 5, 3, 1, 3, 2, 6, 2, 5, 1, 3, 6, 2

Offset: 1

Paul D. Hanna, Aug 27 2005

nonn

			G.f.: Let A(x) = x + 6*x^2 + 6*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + ...
then A(x) = B(B(B(B(B(B(x)))))) where B(x) = x + x^2 - 4*x^3 + 28*x^4 - 236*x^5 + 2159*x^6 + ... is the g.f. of A112113.

Cf. A112113, A112104-A112111, A112114-A112127.

{a(n,m=6)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}

cp's OEIS Frontend

A112127 G.f. A(x) satisfies A(A(A(..(A(x))..))) = B(x) (13th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,..,13}, with B(0) = 0.

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A112105 G.f. A(x) satisfies A(A(x)) = B(x) such that the coefficients of B(x) consist of all 1's and 2's, with A(0) = 0.

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A112117 G.f. A(x) satisfies A(A(A(..(A(x))..))) = B(x) (8th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,..,8}, with B(0) = 0.

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A112106 Unique sequence of numbers {1,2,3} where g.f. A(x) satisfies A(x) = B(B(B(x))) (3rd self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

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A112107 G.f. A(x) satisfies A(A(A(x))) = B(x) (3rd self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3}, with B(0) = 0.

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A112108 Unique sequence of numbers {1,2,3,4} where g.f. A(x) satisfies A(x) = B(B(B(B(x)))) (4th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

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A112109 G.f. A(x) satisfies A(A(A(A(x)))) = B(x) (4th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,4}, with B(0) = 0.

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A112110 Unique sequence of numbers {1,2,3,4,5} where g.f. A(x) satisfies A(x) = B(B(B(B(B(x))))) (5th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

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A112111 G.f. A(x) satisfies A(A(A(A(A(x))))) = B(x) (5th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,4,5}, with B(0) = 0.

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A112112 Unique sequence of numbers {1,2,3,...,6} where g.f. A(x) satisfies A(x) = B(B(B(..(B(x))..))) (6th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

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