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 1-6 of 6 results.

A193040 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A131507(n), where A131507 is defined as "2*n+1 appears n times.".

Original entry on oeis.org

1, 1, 2, 7, 29, 129, 600, 2889, 14293, 72228, 371208, 1934236, 10194853, 54258010, 291175463, 1573878211, 8560931357, 46825444031, 257386132988, 1421034475176, 7876770462043, 43817869686744, 244552276036950, 1368945007588648, 7683977372121530
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2011

Keywords

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 129*x^5 + 600*x^6 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^3 + x^2*A(-x)^3 + x^3*A(-x)^5 + x^4*A(-x)^5 + x^5*A(-x)^5 + x^6*A(-x)^7 +...+ x^n*A(-x)^A131507(n) +...
where A131507 begins: [1,3,3,5,5,5,7,7,7,7,9,9,9,9,9,11,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^2)*A(-x)^3 + x^3*(1-x^3)*A(-x)^5 + x^6*(1-x^4)*A(-x)^7 + x^10*(1-x^5)*A(-x)^9 +...

Crossrefs

Cf. A193039, A193037, A192455, A193050, A131507.

Programs

  • PARI
    {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^(2*floor(sqrt(2*m)+1/2)-1) ),#A));if(n<0,0,A[n+1])}

Formula

G.f. satisfies: 1-x = Sum_{n>=1} x^(n*(n-1)/2)* (1-x^n)* A(-x)^(2*n-1).

A193037 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A022998(n), where A022998 is defined as "if n is odd then n else 2*n.".

Original entry on oeis.org

1, 1, 3, 16, 99, 660, 4625, 33609, 251024, 1915365, 14866307, 117007587, 931682106, 7491746385, 60750081839, 496214311987, 4078991375519, 33718664525501, 280123674031062, 2337556609209193, 19584517345276853, 164677962557101656, 1389268739557153255
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2011

Keywords

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 99*x^4 + 660*x^5 + 4625*x^6 +...
The g.f. satisfies:
x = x*A(-x) + x^2*A(-x)^4 + x^3*A(-x)^3 + x^4*A(-x)^8 + x^5*A(-x)^5 + x^6*A(-x)^12 +...+ x^n*A(-x)^A022998(n) +...
where A022998 begins: [1,4,3,8,5,12,7,16,9,20,11,24,13,28,15,32,...].

Crossrefs

Cf. A193036, A193038, A193039, A193040, A022998.

Programs

  • Mathematica
    CoefficientList[Series[Root[1 - #1 - x^2*#1^2 + (x - x^2)*#1^4 + x^2*#1^5 + (-x^3 + x^4)*#1^6 &, 1], {x, 0, 25}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
  • PARI
    {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^((2-m%2)*m)),#A));if(n<0,0,A[n+1])}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^4-x^2*(A^2+A^4-A^5)-x^3*A^6+x^4*A^6+x*O(x^n));polcoeff(A,n)}

Formula

G.f. satisfies: 1 = A(x)/(1 - x^2*A(x)^2) - x*A(x)^4/(1 - x^2*A(x)^4).
G.f. satisfies: A(x) = 1 - x^2*A(x)^2 + x*(1-x)*A(x)^4 + x^2*A(x)^5 - x^3*(1-x)*A(x)^6.
a(n) ~ sqrt((s^4 - 3*r^2*s^6 + 4*r^3*s^6 + 2*r*s^2*(-1 - s^2 + s^3)) / (6*s^2 - 15*r^2*s^4 + 15*r^3*s^4 + r*(-1 - 6*s^2 + 10*s^3))) / (2 * sqrt(Pi) * n^(3/2) * r^n), where r = 0.1106746599331611304198664461476598606151090027202... and s = 1.30776993974681499155955325507126073446958968382... are real roots of the system of equations 1 + r^2*s^5 + (-1 + r)*r^3*s^6 = s + r^2*s^2 + (-1 + r)*r*s^4, r^2*s^4*(5 + 6*(-1 + r)*r*s) = 1 + 2*r^2*s + 4*(-1 + r)*r*s^3. - Vaclav Kotesovec, Oct 18 2020

A192455 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A001650(n+1), where A001650 is defined by "n appears n times (n odd).".

Original entry on oeis.org

1, 1, 2, 7, 27, 112, 492, 2249, 10580, 50885, 249067, 1236602, 6212563, 31523293, 161317863, 831615320, 4314659345, 22512421092, 118052038100, 621825506334, 3288597601727, 17455485596492, 92958082866815, 496535775228131, 2659574264906443
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2011

Keywords

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 112*x^5 + 492*x^6 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^3 + x^2*A(-x)^3 + x^3*A(-x)^3 + x^4*A(-x)^5 + x^5*A(-x)^5 + x^6*A(-x)^5 + x^7*A(-x)^5 + x^8*A(-x)^5 + x^9*A(-x)^7 +...+ x^n*A(-x)^A001650(n+1) +...
where A001650 begins: [1, 3,3,3, 5,5,5,5,5, 7,7,7,7,7,7,7, 9,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^3 + x^4*(1-x^5)*A(-x)^5 + x^9*(1-x^7)*A(-x)^7 + x^16*(1-x^9)*A(-x)^9 +...

Crossrefs

Cf. A193039, A193040, A193050, A001650.

Programs

  • PARI
    {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+2*sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}

Formula

G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1-x^(2*n-1)) * A(-x)^(2*n-1).

A193036 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A006519(n), where A006519(n) is the highest power of 2 dividing n.

Original entry on oeis.org

1, 1, 1, 3, 10, 34, 112, 382, 1352, 4884, 17856, 66022, 246764, 930878, 3538788, 13542716, 52133416, 201746212, 784378792, 3062431132, 12001867312, 47197716460, 186187480816, 736582735738, 2921679555340, 11617001425938, 46294191726972, 184866924629832
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2011

Keywords

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1).

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 34*x^5 + 112*x^6 + ...
The g.f. satisfies:
x = x*A(-x) + x^2*A(-x)^2 + x^3*A(-x) + x^4*A(-x)^4 + x^5*A(-x) + x^6*A(-x)^2 + x^7*A(-x) + x^8*A(-x)^8 + x^9*A(-x) + ... + x^n * A(-x)^A006519(n) + ...
where A006519 begins: [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,...].
The g.f. also satisfies:
x = x*A(-x)/(1-x^2) + x^2*A(-x)^2/(1-x^4) + x^4*A(-x)^4/(1-x^8) + x^8*A(-x)^8/(1-x^16) + x^16*A(-x)^16/(1-x^32) + x^32*A(-x)^32/(1-x^64) + ...
Related table.
The table of coefficients in A(-x)^(2^n) / (1 - x^(2*2^n)) begins:
n=0: [1, -1, 2, -4, 12, -38, 124, -420, 1476, -5304,  ...];
n=1: [1, -2, 3, -8, 28, -96, 324, -1124, 4024, -14684,  ...];
n=2: [1, -4, 10, -28, 95, -344, 1244, -4512, 16616, -62072, ...];
n=3: [1, -8, 36, -136, 514, -2008, 7924, -31176, 122495, ...];
n=4: [1, -16, 136, -848, 4500, -22032, 103480, -473520, ...];
n=5: [1, -32, 528, -6048, 54632, -418720, 2855088, ...];
n=6: [1, -64, 2080, -45888, 775120, -10720576, 126777952, ...];
n=7: [1, -128, 8256, -358016, 11750304, -311550592, 6955997376, ...];
...
where x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).

Links

Crossrefs

Cf. A193037, A193038, A193039, A193040, A006519.

Programs

  • PARI
    {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^(2^valuation(m,2))),#A));if(n<0,0,A[n+1])}

Formula

G.f. satisfies: x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).

A193050 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A003059(n+1), where A003059 is defined by "n appears 2n-1 times.".

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 38, 87, 204, 489, 1191, 2938, 7328, 18448, 46809, 119583, 307324, 793965, 2060770, 5371156, 14051901, 36887289, 97131351, 256488187, 679046184, 1802047427, 4792800096, 12773166908, 34106055493, 91228795961, 244427136822, 655900969465
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2011

Keywords

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 38*x^7 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^2 + x^2*A(-x)^2 + x^3*A(-x)^2 + x^4*A(-x)^3 + x^5*A(-x)^3 + x^6*A(-x)^3 + x^7*A(-x)^3 + x^8*A(-x)^3 + x^9*A(-x)^4 +...+ x^n*A(-x)^A003059(n+1) +...
where A003059 begins: [1, 2,2,2, 3,3,3,3,3, 4,4,4,4,4,4,4, 5,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^2 + x^4*(1-x^5)*A(-x)^3 + x^9*(1-x^7)*A(-x)^4 + x^16*(1-x^9)*A(-x)^5 +...

Crossrefs

Cf. A192455, A193039, A193040, A003059.

Programs

  • PARI
    {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}

Formula

G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1 - x^(2*n-1)) * A(-x)^n.

A193038 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^sigma(n), where sigma(n) = sum of divisors of n (A000203).

Original entry on oeis.org

1, 1, 2, 6, 23, 101, 475, 2321, 11629, 59364, 307648, 1614724, 8567810, 45890927, 247817187, 1347819147, 7376472346, 40594360200, 224500075274, 1247028876157, 6954322550810, 38921347036195, 218541728743211, 1230754878156173, 6950114772716368
Offset: 0

Views

Author

Paul D. Hanna, Jul 14 2011

Keywords

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 101*x^5 + 475*x^6 +...
The g.f. satisfies:
x = x*A(-x) + x^2*A(-x)^3 + x^3*A(-x)^4 + x^4*A(-x)^7 + x^5*A(-x)^6 + x^6*A(-x)^12 +...+ x^n*A(-x)^A000203(n) +...
where A000203 begins: [1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,...].

Crossrefs

Cf. A193036, A193037, A193039, A193040, A000203.

Programs

  • PARI
    {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^sigma(m)),#A));if(n<0,0,A[n+1])}
Showing 1-6 of 6 results.

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