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 1-3 of 3 results.

A208961 G.f. satisfies: A(x) = 1 + x*[d/dx x/A(x)^2].

Original entry on oeis.org

1, 1, -4, 33, -376, 5255, -85392, 1566656, -31869104, 710089551, -17178977940, 448256023501, -12548355934560, 375195009917364, -11936772609109600, 402740733371490540, -14367278506882083936, 540452504929440595503, -21384560213508955184172
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2012

Keywords

Examples

			G.f.: A(x) = 1 + x - 4*x^2 + 33*x^3 - 376*x^4 + 5255*x^5 - 85392*x^6 +...
where
1/A(x)^2 = 1 - 2*x + 11*x^2 - 94*x^3 + 1051*x^4 - 14232*x^5 +...
The coefficients in A(x)^n begin:
n=1: [1, 1,  -4, 33,  -376,  5255,  -85392, 1566656, ...];
n=2: [1, 2,  -7, 58,  -670,  9494, -156177, 2895672, ...];
n=3: [1, 3,  -9, 76,  -894, 12864, -214339, 4016688, ...];
n=4: [1, 4,(-10),88, -1059, 15496, -261634, 4956000, ...];
n=5: [1, 5,(-10),95, -1175, 17506, -299610, 5736885, ...];
n=6: [1, 6,  -9,(98),-1251, 18996, -329626, 6379902, ...];
n=7: [1, 7,  -7,(98),-1295, 20055, -352870, 6903170, ...];
n=8: [1, 8,  -4, 96,(-1314),20760, -370376, 7322624, ...];
n=9: [1, 9,   0, 93,(-1314),21177, -383040, 7652250, ...];
n=10:[1,10,   5, 90, -1300,(21362),-391635, 7904300, ...];
n=11:[1,11,  11, 88, -1276,(21362),-396825, 8089488, ...];
n=12:[1,12,  18, 88, -1245, 21216,(-399178),8217168, ...];
n=13:[1,13,  26, 91, -1209, 20956,(-399178),8295495, ...];
n=14:[1,14,  35, 98, -1169, 20608, -397236,(8331570), ...];
n=15:[1,15,  45, 110,-1125, 20193, -393700,(8331570), ...]; ...
where the coefficients in parenthesis demonstrate the property:
[x^n] A(x)^(2*n) = [x^n] A(x)^(2*n+1) for n>=2.

Links

Crossrefs

Cf. A185971.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*deriv(x/A^2));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) satisfies: [x^n] A(x)^(2*n) = [x^n] A(x)^(2*n+1) for n>=2.
a(n) ~ c * (-1)^(n+1) * n! * 2^n * n^(3/2), where c = 0.18828692660370683384... - Vaclav Kotesovec, Feb 22 2014

Extensions

Typo in name corrected by Vaclav Kotesovec, Feb 22 2014

A208975 G.f. satisfies: A(x) = 1 + x*A(x) * [d/dx x/A(x)^2].

Original entry on oeis.org

1, 1, -3, 20, -189, 2232, -31130, 497016, -8907885, 176829104, -3849436062, 91187523000, -2335691914050, 64344487654800, -1897619527612692, 59667237154623280, -1993022006345620605, 70488571028815935072, -2631925423768158446390
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2012

Keywords

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + 20*x^3 - 189*x^4 + 2232*x^5 - 31130*x^6 +...
Related expansion:
d/dx x/A(x)^2 = 1 - 4*x + 27*x^2 - 248*x^3 + 2830*x^4 - 38232*x^5 +...
Let G(x) be the g.f. of A000699:
G(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 + 38232*x^7 +...
then A(x) = -x/G(-x), or A(x) = 1 + x*A(x) * (x + G(-x))/x^2.
The coefficients in A(x)^n begin:
n=1: [1, 1, -3, 20, -189, 2232,  -31130,  497016,  -8907885, ...];
n=2: [1, 2, -5, 34, -329, 3966,  -56262,  910820, -16509957, ...];
n=3: [1, 3,(-6),43, -429, 5289,  -76350, 1253250, -22971165, ...];
n=4: [1, 4,(-6),48, -497, 6276,  -92214, 1534560, -28436085, ...];
n=5: [1, 5, -5,(50),-540, 6991, -104555, 1763610, -33031335, ...];
n=6: [1, 6, -3,(50),-564, 7488, -113969, 1948038, -36867735, ...];
n=7: [1, 7,  0, 49,(-574),7812, -120960, 2094415, -40042233, ...];
n=8: [1, 8,  4, 48,(-574),8000, -125952, 2208384, -42639617, ...];
n=9: [1, 9,  9, 48, -567,(8082),-129300, 2294784, -44734032, ...];
n=10:[1,10, 15, 50, -555,(8082),-131300, 2357760, -46390320, ...];
n=11:[1,11, 22, 55, -539, 8019,(-132198),2400860, -47665200, ...];
n=12:[1,12, 30, 64, -519, 7908,(-132198),2427120, -48608304, ...];
n=13:[1,13, 39, 78, -494, 7761, -131469,(2439138),-49263084, ...];
n=14:[1,14, 49, 98, -462, 7588, -130151,(2439138),-49667604, ...];
n=15:[1,15, 60,125, -420, 7398, -128360, 2429025,(-49855230), ...];
n=16:[1,16, 72,160, -364, 7200, -126192, 2410432,(-49855230), ...];
where the coefficients in parenthesis demonstrate the property:
[x^n] A(x)^(2*n-1) = [x^n] A(x)^(2*n) for n>=2.

Links

Crossrefs

Cf. A185971, A208961, A000699.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*A*deriv(x/A^2));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) satisfies: [x^n] A(x)^(2*n-1) = [x^n] A(x)^(2*n) for n>=2.
G.f.: A(x) = -x/G(-x) where G(x) is the g.f. of A000699, the number of irreducible diagrams with 2n nodes.
a(n) ~ -(-1)^n * 2^(n + 3/2) * n^(n+1) / exp(n+1). - Vaclav Kotesovec, Nov 18 2017

A295553 Expansion of 1/(1 - Sum_{k>=1} (2*k-1)!!*x^k).

Original entry on oeis.org

1, 1, 4, 22, 154, 1330, 13882, 171802, 2474098, 40738594, 755322778, 15566915770, 352862768434, 8720662458754, 233285616212506, 6713983428179098, 206813607458357746, 6788092999359053410, 236481982146071359258, 8714521818620631672058, 338660320676350494328882, 13841377309645038610883266
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2017

Keywords

Comments

Invert transform of A001147.
Number of compositions (ordered partitions) of n where there are 1*3*5*...*(2*k-1) sorts of part k.

Links

Crossrefs

Cf. A001147, A051296, A185971, A292778.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - Sum[(2 k - 1)!! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[1 + x/(1 - 2 x + ContinuedFractionK[-k x, 1, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(2 k - 1)!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A001147(k)*x^k).
G.f.: 1 + x/(1 - 2*x - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction.
a(0) = 1; a(n) = Sum_{k=1..n} (2*k-1)!!*a(n-k).
Showing 1-3 of 3 results.

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