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 1-10 of 40 results. Next

A384619 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A213591.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 24, 0, 1, 4, 15, 56, 178, 0, 1, 5, 22, 97, 420, 1512, 0, 1, 6, 30, 148, 738, 3572, 14152, 0, 1, 7, 39, 210, 1145, 6300, 33328, 142705, 0, 1, 8, 49, 284, 1655, 9832, 58702, 334354, 1528212, 0, 1, 9, 60, 371, 2283, 14321, 91640, 586635, 3559310, 17211564, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2025

Keywords

Examples

			Square array begins:
  1,     1,     1,     1,     1,      1,      1, ...
  0,     1,     2,     3,     4,      5,      6, ...
  0,     4,     9,    15,    22,     30,     39, ...
  0,    24,    56,    97,   148,    210,    284, ...
  0,   178,   420,   738,  1145,   1655,   2283, ...
  0,  1512,  3572,  6300,  9832,  14321,  19938, ...
  0, 14152, 33328, 58702, 91640, 133720, 186753, ...

Crossrefs

Columns k=0..1 give A000007, A213591(n+1).
Cf. A379599, A384620, A384621, A384623.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*a(n-j, 2*j)));

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * A(n-j,2*j).

A275765 G.f. satisfies: A(x - A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 2, 12, 106, 1148, 14156, 191400, 2775930, 42585412, 684496988, 11449962008, 198331811356, 3543990791480, 65136985937096, 1228531761076208, 23733123786608826, 468887742020767788, 9461919438245032500, 194817077269127033944, 4089069139317823277548, 87426000975842460304792, 1902787414323673070857528, 42133267254272433484761584, 948695717599714654940068604, 21712101305047777916075831096, 504865916349551192319293625016
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1148*x^5 + 14156*x^6 + 191400*x^7 + 2775930*x^8 + 42585412*x^9 + 684496988*x^10 + 11449962008*x^11 + 198331811356*x^12 +...
such that A(x - A(x)^2) = x + A(x)^2.
RELATED SERIES.
Series_Reversion(x - A(x)^2) = x + x^2 + 6*x^3 + 53*x^4 + 574*x^5 + 7078*x^6 + 95700*x^7 + 1387965*x^8 + 21292706*x^9 + 342248494*x^10 +...
which equals (A(x) + x)/2.
A( (A(x) + x)/2 ) = x + 3*x^2 + 22*x^3 + 221*x^4 + 2634*x^5 + 35086*x^6 + 506356*x^7 + 7773279*x^8 + 125441594*x^9 + 2110832382*x^10 +...
which equals sqrt( (A(x) - x)/2 ).
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 2*x^2 - 4*x^3 - 26*x^4 - 228*x^5 - 2396*x^6 - 28440*x^7 - 369114*x^8 - 5135468*x^9 - 75602108*x^10 - 1167066216*x^11 - 18768202924*x^12 +...
then Series_Reversion(x + A(x)^2) = x/2 + R(x)/2.

Links

Crossrefs

Cf. A277295, A213591, A276364, A276360, A276361, A276362, A276363.

Programs

  • Mathematica
    m = 26; A[_] = 0;
Do[A[x_] = x + 2 A[x/2 + A[x]/2]^2 + O[x]^(m+1) // Normal, {m+1}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = -polcoeff(subst(F,x,x-F^2) - F^2,#A) );A[n]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 2 * A( x/2 + A(x)/2 )^2.
(2) A(x) = -x + 2 * Series_Reversion(x - A(x)^2).
(3) R(x) = -x + 2 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/2 - R(x)/2 ) ) = x/2 + R(x)/2, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^(n-k).

A277295 G.f. A(x,y) satisfies: A( x - y*A(x,y)^2, y) = x + (1-y)*A(x,y)^2, where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 5, 14, 5, 0, 14, 74, 76, 14, 0, 42, 352, 698, 378, 42, 0, 132, 1588, 5088, 5404, 1808, 132, 0, 429, 6946, 32461, 56410, 37546, 8484, 429, 0, 1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0, 4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0, 16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0, 58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0
Offset: 1

Views

Author

Paul D. Hanna, Oct 11 2016

Keywords

Comments

More generally, we have the following related identity.
Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,
if F(x - y*G(x)) = x + (1-y)*G(x), then
(1) F(x) = x + G( y*F(x) + (1-y)*x ),
(2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),
(3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),
(4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.
The g.f. of this sequence A(x,y) equals F(x) in the above when G(x) = F(x)^2.

Examples

			G.f.: A(x,y)  = x + x^2 + (2*y + 2)*x^3 + (5*y^2 + 14*y + 5)*x^4 + (14*y^3 + 76*y^2 + 74*y + 14)*x^5 + (42*y^4 + 378*y^3 + 698*y^2 + 352*y + 42)*x^6 + (132*y^5 + 1808*y^4 + 5404*y^3 + 5088*y^2 + 1588*y + 132)*x^7 + (429*y^6 + 8484*y^5 + 37546*y^4 + 56410*y^3 + 32461*y^2 + 6946*y + 429)*x^8 + (1430*y^7 + 39446*y^6 + 244220*y^5 + 535410*y^4 + 486550*y^3 + 189940*y^2 + 29786*y + 1430)*x^9 + (4862*y^8 + 182732*y^7 + 1522466*y^6 + 4597402*y^5 + 6036632*y^4 + 3690410*y^3 + 1046190*y^2 + 126008*y + 4862)*x^10 +...
such that
A( x - y*A(x,y)^2, y)  =  x + (1-y)*A(x,y)^2.
Also,
A(x,y) = x + A( y*A(x,y) + (1-y)*x, y)^2.
...
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:
1;
1, 0;
2, 2, 0;
5, 14, 5, 0;
14, 74, 76, 14, 0;
42, 352, 698, 378, 42, 0;
132, 1588, 5088, 5404, 1808, 132, 0;
429, 6946, 32461, 56410, 37546, 8484, 429, 0;
1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0;
4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0;
16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0;
58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0; ...
RELATED SEQUENCES.
Given T(n,k) is the coefficient of x^n*y^k in g.f. A(x,y),
if b(n) = Sum_{k=0..n-1} T(n,k) * p^k * q^(n-k-1)
then B(x) = Sum_{n>=1} b(n)*x^n satisfies
(1) B(x - p*B(x)^2) = x + (q-p)*B(x)^2
(2) B(x)  =  x + B( p*B(x) + (q-p)*x )^2.
Examples:
A213591(n) = sum(k=0,n-1, T(n,k) )
A275765(n) = sum(k=0,n-1, T(n,k) * 2^(n-k) )
A276360(n) = sum(k=0,n-1, T(n,k) * 3^(n-k-1) )
A276361(n) = sum(k=0,n-1, T(n,k) * 2^k * 3^(n-k-1) )
A276362(n) = sum(k=0,n-1, T(n,k) * 4^(n-k-1) )
A276363(n) = sum(k=0,n-1, T(n,k) * 3^k * 4^(n-k-1) )
A276365(n) = sum(k=0,n-1, T(n,k) * 2^k )
A277300(n) = sum(k=0,n-1, T(n,k) * 5^(n-k-1) )
A277301(n) = sum(k=0,n-1, T(n,k) * 2^k * 5^(n-k-1) )
A277302(n) = sum(k=0,n-1, T(n,k) * 3^k * 5^(n-k-1) )
A277303(n) = sum(k=0,n-1, T(n,k) * 4^k * 5^(n-k-1) )
A277304(n) = sum(k=0,n-1, T(n,k) * 6^(n-k-1) )
A277305(n) = sum(k=0,n-1, T(n,k) * 5^k * 6^(n-k-1) )
A277306(n) = sum(k=0,n-1, T(n,k) * (-1)^k )
A277307(n) = sum(k=0,n-1, T(n,k) * 3^k )
A277308(n) = sum(k=0,n-1, T(n,k) * 3^k * 2^(n-k-1) )
A277309(n) = sum(k=0,n-1, T(n,k) * 5^k * 2^(n-k-1) )
A277310(n) = sum(k=0,n-1, T(n,k) * 4^k )
A277311(n) = sum(k=0,n-1, T(n,k) * 5^k )
...

Links

Crossrefs

Cf. A000108 (column 0), A138156 (column 1), A277296 (column 2), A277297 (diagonal), A277298 (central terms T(2*n-1,n-1)), A277299 (central terms T(2*n,n-1)).
Cf. A213591 (row sums), A275765, A276360, A276361, A276362, A276363, A276365.
Cf. A277300, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309, A277310, A277311.

Programs

  • Mathematica
    c[n_] := c[n] = Module[{A}, A[x_] = x; Do[A[x_] = x + A[y A[x] + (1-y) x + x O[x]^j]^2, {j, n}] // Normal; SeriesCoefficient[A[x], {x, 0, n}] // Expand];
T[n_, k_] := SeriesCoefficient[c[n], {y, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {T(n,k) = my(A=x); for(i=1, n, A = x + subst(A^2, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A,n,x),k,y)}
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))

Formula

G.f. A(x,y) also satisfies:
(1) A(x,y) = x + A( y*A(x,y) + (1-y)*x, y)^2.
(2) y*A(x,y) + (1-y)*x = Series_Reversion( x - y*A(x,y)^2 ).
(3) y*x + (1-y)*B(x,y) = Series_Reversion( x + (1-y)*A(x,y)^2 ), where B( A(x,y), y) = x.
(4) A(x,y) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) A(x,y)^(2*n) / n!.
In formulas 2 and 3, the series reversion is taken with respect to variable x.
T(n+1,0) = T(n+1,n-1) = binomial(2*n,n)/(n+1) = A000108(n) for n>=1.
T(n+1,1) = 4^n - (3*n+1)*binomial(2*n,n)/(n+1) = A138156(n-1) for n>=1.

A276360 G.f. satisfies: A(x - A(x)^2) = x + 2*A(x)^2.

Original entry on oeis.org

1, 3, 24, 276, 3858, 61092, 1056816, 19550475, 381543576, 7782820548, 164842646424, 3607654164924, 81281990795520, 1879865970374568, 44527769989124976, 1078220967132218616, 26650484274297181896, 671558570413109457264, 17234310756238557856200, 450044549619831325213920, 11949386806898017225833312, 322394088574898542428753168, 8833647058171126097908059720
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 24*x^3 + 276*x^4 + 3858*x^5 + 61092*x^6 + 1056816*x^7 + 19550475*x^8 + 381543576*x^9 + 7782820548*x^10 + 164842646424*x^11 + 3607654164924*x^12 +...
such that A(x - A(x)^2) = x + 2*A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - A(x)^2) = 2*x/3 + A(x)/3, which begins:
Series_Reversion(x - A(x)^2) = x + x^2 + 8*x^3 + 92*x^4 + 1286*x^5 + 20364*x^6 + 352272*x^7 + 6516825*x^8 + 127181192*x^9 + 2594273516*x^10 + 54947548808*x^11 +
1202551388308*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 3*x^2 - 6*x^3 - 51*x^4 - 564*x^5 - 7416*x^6 - 109764*x^7 - 1772028*x^8 - 30603930*x^9 - 558238326*x^10 - 10659285096*x^11 - 211688430204*x^12 +...
then Series_Reversion(x + 2*A(x)^2) = x/3 + 2*R(x)/3.

Links

Crossrefs

Cf. A277295, A213591, A275765, A276361, A276362, A276363.

Programs

  • Mathematica
    m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[2 x/3 + A[x]/3]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^2) - 2*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 3 * A( 2*x/3 + A(x)/3 )^2.
(2) A(x) = -2*x + 3 * Series_Reversion(x - A(x)^2).
(3) 2*R(x) = -x + 3 * Series_Reversion(x + 2*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/3 - R(x)/3 ) ) = x/3 + 2*R(x)/3, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*3^(n-k-1).

A276361 G.f. satisfies: A(x - 2*A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 3, 30, 447, 8202, 171846, 3956796, 97916895, 2567551890, 70655670690, 2026596875268, 60282027684678, 1852444347792036, 58633762133405100, 1907098496516434680, 63620675921801106495, 2173457638433471757282, 75940916632597398212298, 2710857429948875567968692, 98775527832178103444182722, 3670845430153146908693608044, 139047871842184594320103381524, 5365224711989826990651317756232
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 30*x^3 + 447*x^4 + 8202*x^5 + 171846*x^6 + 3956796*x^7 + 97916895*x^8 + 2567551890*x^9 + 70655670690*x^10 + 2026596875268*x^11 + 60282027684678*x^12 +...
such that A(x - 2*A(x)^2) = x + A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 2*A(x)^2) = x/3 + 2*A(x)/3, which begins:
Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 20*x^3 + 298*x^4 + 5468*x^5 + 114564*x^6 + 2637864*x^7 + 65277930*x^8 + 1711701260*x^9 + 47103780460*x^10 + 1351064583512*x^11 + 40188018456452*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 3*x^2 - 12*x^3 - 132*x^4 - 1992*x^5 - 36144*x^6 - 742176*x^7 - 16688880*x^8 - 402824928*x^9 - 10300868160*x^10 - 276531035520*x^11 - 7742210941056*x^12 +...
then Series_Reversion(x + A(x)^2) = 2*x/3 + R(x)/3.

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276362, A276363.

Programs

  • Mathematica
    m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[x/3 + 2 A[x]/3]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-2*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 3 * A( x/3 + 2*A(x)/3 )^2.
(2) 2*A(x) = -x + 3 * Series_Reversion(x - 2*A(x)^2).
(3) R(x) = -2*x + 3 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/3 - R(x)/3 ) ) = 2*x/3 + R(x)/3, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k*3^(n-k-1).

A276362 G.f. satisfies: A(x - A(x)^2) = x + 3*A(x)^2.

Original entry on oeis.org

1, 4, 40, 564, 9592, 184008, 3844624, 85700980, 2011283640, 49248127800, 1250064156912, 32736194249256, 881252194701616, 24317581366876880, 686300288661644960, 19774058901706750100, 580795172081872246232, 17368587281321383296184, 528294942152813411073968, 16329939570298980826852824, 512590568042639978453793744, 16329084800479729420462546352, 527621994750854274463428080608
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9592*x^5 + 184008*x^6 + 3844624*x^7 + 85700980*x^8 + 2011283640*x^9 + 49248127800*x^10 + 1250064156912*x^11 + 32736194249256*x^12 +...
such that A(x - A(x)^2) = x + 3*A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - A(x)^2) = 3*x/4 + A(x)/4, which begins:
Series_Reversion(x - A(x)^2) = x + x^2 + 10*x^3 + 141*x^4 + 2398*x^5 + 46002*x^6 + 961156*x^7 + 21425245*x^8 + 502820910*x^9 + 12312031950*x^10 + 312516039228*x^11 + 8184048562314*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 4*x^2 - 8*x^3 - 84*x^4 - 1112*x^5 - 17352*x^6 - 303824*x^7 - 5791060*x^8 - 117898648*x^9 - 2531645240*x^10 - 56835852080*x^11 - 1325547044072*x^12 +...
then Series_Reversion(x + 3*A(x)^2) = x/4 + 3*R(x)/4.

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276363.

Programs

  • Mathematica
    m = 24; A[_] = 0;
Do[A[x_] = x + 4 A[3x/4  + A[x]/4]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^2) - 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 4 * A( 3*x/4 + A(x)/4 )^2.
(2) A(x) = -3*x + 4 * Series_Reversion(x - A(x)^2).
(3) 3*R(x) = -x + 4 * Series_Reversion(x + 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/3 - R(x)/3 ) ) = x/4 + 3*R(x)/4, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*4^(n-k-1).

A276363 G.f. satisfies: A(x - 3*A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 4, 56, 1172, 30248, 892296, 28951344, 1010322900, 37384819496, 1452697058744, 58872642043856, 2475764515398568, 107619880380347920, 4821324372637921744, 222077355203506939104, 10497354682052048593332, 508414637258604924680136, 25197644191294099697736312, 1276547957544912412461457680, 66046883289153773427379134360, 3487101507192780951408327918192
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 4*x^2 + 56*x^3 + 1172*x^4 + 30248*x^5 + 892296*x^6 + 28951344*x^7 + 1010322900*x^8 + 37384819496*x^9 + 1452697058744*x^10 + 58872642043856*x^11 +
2475764515398568*x^12 +...
such that A(x - 3*A(x)^2) = x + A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 3*A(x)^2) = x/4 + 3*A(x)/4, which begins:
Series_Reversion(x - 3*A(x)^2) = x + 3*x^2 + 42*x^3 + 879*x^4 + 22686*x^5 + 669222*x^6 + 21713508*x^7 + 757742175*x^8 + 28038614622*x^9 + 1089522794058*x^10 + 44154481532892*x^11 + 1856823386548926*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 4*x^2 - 24*x^3 - 372*x^4 - 7944*x^5 - 204168*x^6 - 5942256*x^7 - 189500916*x^8 - 6490281480*x^9 - 235609789368*x^10 - 8983294304784*x^11 - 357373688297448*x^12 +...
then Series_Reversion(x + A(x)^2) = 3*x/4 + R(x)/4.

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362.

Programs

  • Mathematica
    m = 22; A[_] = 0;
Do[A[x_] = x + 4A[x/4 + 3A[x]/4]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 3*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 4 * A( x/4 + 3*A(x)/4 )^2.
(2) 3*A(x) = -x + 4 * Series_Reversion(x - 3*A(x)^2).
(3) R(x) = -3*x + 4 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/4 - R(x)/4 ) ) = 3*x/4 + R(x)/4, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*3^k*4^(n-k-1).

A139702 G.f. satisfies: x = A( x + A(x)^2 ).

Original entry on oeis.org

1, -1, 4, -24, 178, -1512, 14152, -142705, 1528212, -17211564, 202460400, -2474708496, 31310415376, -408815254832, 5495451727376, -75907303147652, 1075685334980240, -15618612118252960, 232102241507321384, -3526880759915999016
Offset: 1

Views

Author

Paul D. Hanna, Apr 30 2008, May 20 2008

Keywords

Comments

Signed version of A213591.

Examples

			G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
where A(x + A(x)^2) = x.
Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then:
G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and
G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+...
so that G(x) = G(G(x)) - x^2 = g.f. of A138740.
Logarithmic series:
log(A(x)/x) = -A(x)^2/x + [d/dx A(x)^4/x]/2! - [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! -+...

Links

Crossrefs

Cf. A138740, A212411, A213591.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.

Programs

  • Mathematica
    nmax = 20; sol = {a[1] -> 1}; nmin = Length[sol]+1;
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[x - A[x + A[x]^2] + O[x]^(n+1), x][[nmin;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nmin, nmax}];
a /@ Range[nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(A, n))}
  • PARI
    /* n-th Derivative: */
{Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{a(n)=local(A=x-x^2+x*O(x^n));for(i=1,n,
A=x*exp(sum(m=0,n,(-1)^(m+1)*Dx(m,A^(2*m+2)/x)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}

Formula

Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740.
G.f. satisfies: A(x) = x*G(-A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 - x*B^2;
B = A - x*C^2;
C = B - x*D^2;
D = C - x*E^2;
E = D - x*F^2; ...
G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [Paul D. Hanna, Dec 18 2010]

A277300 G.f. satisfies: A(x - A(x)^2) = x + 4*A(x)^2.

Original entry on oeis.org

1, 5, 60, 1000, 19970, 448160, 10926360, 283651245, 7740058300, 220046970860, 6476695275680, 196438030797880, 6117627849485360, 195082685133612800, 6355848358118392400, 211189970909192038500, 7146354688384980282000, 245970478274041025623200, 8602606263466490521359400, 305460999044315834902424200, 11003870605124169641012461600
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 19970*x^5 + 448160*x^6 + 10926360*x^7 + 283651245*x^8 + 7740058300*x^9 + 220046970860*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • Mathematica
    m = 22; A[_] = 0;
Do[A[x_] = x + 5 A[4x/5 + A[x]/5]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^2) - 4*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 5 * A( 4*x/5 + A(x)/5 )^2.
(2) A(x) = -4*x + 5 * Series_Reversion(x - A(x)^2).
(3) R(x) = -x/4 + 5/4 * Series_Reversion(x + 4*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = x/5 + 4*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^(n-k-1).

A277301 G.f. satisfies: A(x - 2*A(x)^2) = x + 3*A(x)^2.

Original entry on oeis.org

1, 5, 70, 1425, 35410, 999210, 30855820, 1020407105, 35642665050, 1302725802510, 49490450201460, 1944619121474970, 78734794663758580, 3275324221277662900, 139667810517388712600, 6093781146211490413825, 271623891311306597652650, 12353670814537544856558950, 572686428900679117724156900, 27036308383662996662940155550, 1298856469077709523772645582300
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 70*x^3 + 1425*x^4 + 35410*x^5 + 999210*x^6 + 30855820*x^7 + 1020407105*x^8 + 35642665050*x^9 + 1302725802510*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 2*F^2) - 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 5 * A( 3*x/5 + 2*A(x)/5 )^2.
(2) A(x) = -3*x/2 + 5/2 * Series_Reversion(x - 2*A(x)^2).
(3) R(x) = -2*x/3 + 5/3 * Series_Reversion(x + 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = 2*x/5 + 3*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 2^k * 5^(n-k-1).
Showing 1-10 of 40 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.