A084212 -id:A084212 - cp's OEIS frontend

A084202 G.f. A(x) defined by: A(x)^2 consists entirely of integer coefficients between 1 and 2 (A083952); A(x) is the unique power series solution with A(0)=1.

1, 1, 0, 1, 0, 1, -1, 2, -2, 4, -6, 10, -16, 27, -44, 75, -127, 218, -375, 650, -1130, 1974, -3460, 6086, -10736, 18993, -33685, 59882, -106683, 190446, -340611, 610243, -1095102, 1968200, -3542468, 6384518, -11521308, 20815942, -37651528, 68176596, -123574852, 224204708, -407153894

Offset: 0

Paul D. Hanna, May 19 2003

sign

Limit a(n)/a(n+1) -> r = -0.530852489019085 where A(r)=0.

Let A_n(x) be the power series formed from the first n terms of this sequence. Then a(0) = 1, a(n) = floor(1 - [x^n] (A_(n-1)(x))^2/2). Replacing 2 with a larger integer k generates the related sequences A084203-A084212. - Charlie Neder, Jan 16 2019

Paul D. Hanna, Table of n, a(n) for n = 0..1024
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083952, A084203-A084212.

a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Table[a[n], {n, 0, 42}]; CoefficientList[ Series[ Sqrt[ Sum[ a[i]*x^i, {i, 0, 42}]], {x, 0, 42}], x] (* Robert G. Wilson v, Nov 11 2007 *)

/* Using Charlie Neder's formula */
{a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = floor(1 - polcoeff( Ser(A)^2, #A-1)/2) ); A[n+1]}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Jan 17 2019

A084203 G.f. A(x) defined by: A(x)^3 consists entirely of integer coefficients between 1 and 3 (A083953); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, 0, 0, 1, -1, 2, -2, 2, 0, -4, 12, -24, 38, -46, 33, 29, -176, 443, -827, 1222, -1310, 433, 2488, -8814, 19528, -33599, 44928, -37805, -17916, 168049, -463252, 921694, -1446018, 1679053, -808620, -2598482, 10515127, -24690122, 44515322, -62719429, 58496244, 10670109, -213311788, 632128236

Offset: 0

Paul D. Hanna, May 19 2003

sign

Does limit_{n ->infinity} a(n)/a(n+1) exist?

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083953, A084202, A084204-A084212.

kmax = 45;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^3 + O[x]^kmax , x];
r = {};
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 3, a[k-1], Integers] // ToRules]; coes = coes /. r, {k, 1, kmax}];
Array[a, kmax, 0] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -1, 3, -7, 20, -58, 177, -554, 1769, -5739, 18866, -62684, 210146, -709882, 2413743, -8253995, 28366316, -97916761, 339326189, -1180068800, 4116957243, -14404398636, 50530280752, -177684095927, 626181400993, -2211215950469, 7823025701314, -27724997048327

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.269562488839799 where A(r)=0.

Robert Israel, Table of n, a(n) for n = 0..1000
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083954, A084202, A084203, A084205-A084212.

g:= 1: a[0]:= 1:
for n from 1 to 50 do
  a[n]:= -floor((coeff(g^4,x,n)-1)/4);
  g:= g + a[n]*x^n;
od:
seq(a[n],n=0..50); # Robert Israel, Sep 04 2019

kmax = 30;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^4 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1}; coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 4, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084205 G.f. A(x) defined by: A(x)^5 consists entirely of integer coefficients between 1 and 5 (A083945); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -1, 3, -8, 24, -76, 252, -854, 2950, -10343, 36706, -131570, 475576, -1731357, 6342042, -23356185, 86421603, -321111661, 1197586539, -4481348585, 16819759474, -63302097780, 238835017492, -903165412289, 3422512973645, -12994514592311, 49425252955926

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.2512525316047635 where A(r)=0.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083945, A084202-A084204, A084206-A084212.

kmax = 30;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^5 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 5, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax+1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084207 G.f. A(x) defined by: A(x)^7 consists entirely of integer coefficients between 1 and 7 (A083947); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -2, 8, -34, 158, -768, 3858, -19851, 104023, -552974, 2973832, -16146688, 88376636, -487034106, 2699839758, -15043262970, 84197804254, -473140314356, 2668221663736, -15095165871964, 85645090974518, -487190919969502

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.166670835025545 where A(r)=0.

Cf. A083947, A084202-A084206, A084208-A084212.

kmax = 25;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^7 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 7, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084211 G.f. A(x) defined by: A(x)^11 consists entirely of integer coefficients between 1 and 11 (A084066); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -4, 26, -189, 1479, -12106, 102224, -883031, 7761209, -69153920, 623018880, -5664270185, 51892998965, -478521450110, 4437418074830, -41350439060725, 386983852716405

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.100000000011 where A(r)=0.

Cf. A084066, A084202-A084210, A084212.

kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^11 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 11, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084206 G.f. A(x) defined by: A(x)^6 consists entirely of integer coefficients between 1 and 6 (A083946); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -2, 7, -27, 115, -510, 2343, -11029, 52896, -257457, 1268098, -6307546, 31633044, -159757597, 811708539, -4145882814, 21273287952, -109603172373, 566748274099, -2940175511195, 15297961574259, -79808998488751, 417373462315834

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.1815238859919 where A(r)=0.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A083946, A084202-A084205, A084207-A084212.

kmax = 25;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^6 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 6, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084208 G.f. A(x) defined by: A(x)^8 consists entirely of integer coefficients between 1 and 8 (A083948); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -3, 15, -82, 484, -2992, 19110, -124979, 832234, -5621028, 38402783, -264858143, 1841221687, -12886279885, 90713376563, -641815393278, 4561172770669, -32542369727538, 232992967457839

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.131401689761435 where A(r)=0.

Cf. A083948, A084202-A084207, A084209-A084212.

kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^8 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 8, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084209 G.f. A(x) defined by: A(x)^9 consists entirely of integer coefficients between 1 and 9 (A083949); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -3, 15, -85, 523, -3367, 22371, -152104, 1052568, -7385756, 52410754, -375382683, 2709626768, -19688989762, 143885743077, -1056748051734, 7795106129504, -57723430872280, 428923406694402

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.126715878986521 where A(r)=0.

Cf. A083949, A084202-A084208, A084210-A084212.

kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^9 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 9, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A084210 G.f. A(x) defined by: A(x)^10 consists entirely of integer coefficients between 1 and 10 (A083950); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -4, 25, -173, 1292, -10105, 81565, -673691, 5662878, -48263038, 415950272, -3617999891, 31714089336, -279828926113, 2483097203637, -22143011361045, 198317403322755

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.104430987675729 where A(r)=0.

Cf. A083950, A084202-A084209, A084211, A084212.

kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^10 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 10, a[k - 1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

cp's OEIS Frontend

A084202 G.f. A(x) defined by: A(x)^2 consists entirely of integer coefficients between 1 and 2 (A083952); A(x) is the unique power series solution with A(0)=1.

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A084203 G.f. A(x) defined by: A(x)^3 consists entirely of integer coefficients between 1 and 3 (A083953); A(x) is the unique power series solution with A(0)=1.

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A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

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A084205 G.f. A(x) defined by: A(x)^5 consists entirely of integer coefficients between 1 and 5 (A083945); A(x) is the unique power series solution with A(0)=1.

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A084207 G.f. A(x) defined by: A(x)^7 consists entirely of integer coefficients between 1 and 7 (A083947); A(x) is the unique power series solution with A(0)=1.

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A084211 G.f. A(x) defined by: A(x)^11 consists entirely of integer coefficients between 1 and 11 (A084066); A(x) is the unique power series solution with A(0)=1.

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A084206 G.f. A(x) defined by: A(x)^6 consists entirely of integer coefficients between 1 and 6 (A083946); A(x) is the unique power series solution with A(0)=1.

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A084208 G.f. A(x) defined by: A(x)^8 consists entirely of integer coefficients between 1 and 8 (A083948); A(x) is the unique power series solution with A(0)=1.

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A084209 G.f. A(x) defined by: A(x)^9 consists entirely of integer coefficients between 1 and 9 (A083949); A(x) is the unique power series solution with A(0)=1.

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