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.

A317352 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).

Original entry on oeis.org

1, 1, 5, 42, 471, 6435, 102232, 1837630, 36719439, 805716679, 19239923577, 496514053880, 13769677836500, 408449335836132, 12906850662570996, 432942515731367894, 15367227978734187567, 575544844737119275935, 22685977410186834271463, 938867118118688412116554
Offset: 0

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Author

Paul D. Hanna, Jul 26 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + 36719439*x^8 + 805716679*x^9 + ...
where A(x) = 1 + x*A(x)^3 + 2*x^2*A(x)^2*A'(x).
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*x*A(x)^2 ) / A(x) begins:
n=1: [1, 0, -5, -158, -7779, -563924, -56177105, -7318104450, ...];
n=2: [1, 1, 0, -94, -5968, -473688, -49352768, -6601523360, ...];
n=3: [1, 2, 7, 0, -3435, -354282, -40709709, -5723430444, ...];
n=4: [1, 3, 16, 130, 0, -199016, -29893568, -4657391616, ...];
n=5: [1, 4, 27, 302, 4541, 0, -16486865, -3372747590, ...];
n=6: [1, 5, 40, 522, 10416, 251976, 0, -1833979680, ...];
n=7: [1, 6, 55, 796, 17877, 567562, 20138467, 0, ...];
n=8: [1, 7, 72, 1130, 27200, 958968, 44592256, 2176638976, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Links

Crossrefs

Cf. A088716, A208961, A317353, A317354, A352236.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = 1 + x*A*(x*A^2)' +x*O(x^n)); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); m=#A-1; A[#A] = -Vec( exp(-m * x*Ser(A)^2 ) * (2 - 1/Ser(A)))[#A] ); A[n+1]}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); m=#A-1; A[#A] = -Vec( exp(-m^2 * x*Ser(A)^2 ) * (m+1 - m/Ser(A)))[#A]/m ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

O.g.f. A(x) satisfies:
(1) A(x) = 1 + x*A(x) * ( d/dx x*A(x)^2 ).
(2) [x^n] exp( n * x*A(x)^2 ) / A(x) = 0 for n>0.
(3.a) [x^n] exp(-n * x*A(x)^2) * (2 - 1/A(x)) = 0 for n >= 1.
(3.b) [x^n] exp(-n^2 * x*A(x)^2) * (n + 1 - n/A(x)) = 0 for n >= 1.
(3.c) [x^n] exp(-n^(p+1) * x*A(x)^2) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.
a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.188286926603706833845600622... - Vaclav Kotesovec, Aug 05 2018

A185971 Convolution inverse of A001147.

Original entry on oeis.org

1, -1, -2, -10, -74, -706, -8162, -110410, -1708394, -29752066, -576037442, -12277827850, -285764591114, -7213364729026, -196316804255522, -5731249477826890, -178676789473121834, -5925085744543837186
Offset: 0

Views

Author

Michael Somos, Feb 08 2011

Keywords

Examples

			1 - x - 2*x^2 - 10*x^3 - 74*x^4 - 706*x^5 - 8162*x^6 - 110410*x^7 - ...
From _Paul D. Hanna_, Mar 03 2012: (Start)
The coefficients in A(x)^n begin:
n=1: [1,  -1, -2, -10,  -74,  -706,  -8162, -110410, ...];
n=2: [1,  -2,(-3),-16, -124, -1224, -14516, -200192, ...];
n=3: [1,  -3,(-3),-19, -156, -1596, -19412, -272772, ...];
n=4: [1,  -4, -2,(-20),-175, -1856, -23136, -331008, ...];
n=5: [1,  -5,  0,(-20),-185, -2031, -25920, -377280, ...];
n=6: [1,  -6,  3, -20,(-189),-2142, -27951, -413568, ...];
n=7: [1,  -7,  7, -21,(-189),-2205, -29379, -441519, ...];
n=8: [1,  -8, 12, -24, -186,(-2232),-30324, -462504, ...];
n=9: [1,  -9, 18, -30, -180,(-2232),-30882, -477666, ...];
n=10:[1, -10, 25, -40, -170, -2212,(-31130),-487960, ...];
n=11:[1, -11, 33, -55, -154, -2178,(-31130),-494186, ...]; ...
where the coefficients in parenthesis demonstrate the properties:
(2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2,
(3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1:
A000699 = [1/1, 3/3, 20/5, 189/7, 2232/9, 31130/11, ...].
Note: g.f. of A000699, G(x), satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].
(End)

Links

Crossrefs

Cf. A001147, A000699, A185971, A208975, A208961.

Programs

  • Mathematica
    a[n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / (Sum[ (2 k - 1)!! x^k, {k, 0, n}] + O[x]^(n + 1)), n]];
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / sum( k=0, n, x^k * (2*k)! / (2^k * k!), x * O(x^n)), n))}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1-x*A^2*deriv(x/A^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2012 */
  • Sage
    def A185971_list(len): # len >= 1
    if len == 1: return [1]
    T = [0]*(2*len-1); T[1] = 1; R = [1,-1]
    for n in (1..2*len-3):
        a,b,c = 1,0,0
        for k in range(n,-1,-1):
            r = a-(k+2)*c
            if k < n : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
        if is_even(n): R.append(-abs(u))
    return R
A185971_list(18)  # Peter Luschny, Nov 01 2012

Formula

G.f.: 1 / ( Sum_{k>=0} (2*k-1)!! * x^k ).
a(n) = -A000698(n) if n > 0.
G.f. A(x) = 1 - x * B(x) * C(x) where B = g.f. for A001147 and C = g.f. for A005416.
G.f.: A(x) = 1 - x/W(0); W(k) = 1 + x + x*2k - x*(2k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
From Paul D. Hanna, Mar 03 2012: (Start)
G.f. A(x) satisfies:
(1) A(x) = 1 - x*A(x)^2 * [d/dx x/A(x)^2].
(2) [x^n] A(x)^(2*n-2) = [x^n] A(x)^(2*n-1) for n>=2.
(3) [x^n] A(x)^(2*n-1) = -(2*n-1)*A000699(n) for n>=1. (End)
G.f. A(x) = G(0) where G(k)= 1 - x*(2*k+1)/(1 - (2*k+2)*x/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 11 2012
G.f. A(x)=1-x/Q(0) where Q(k)= 1 - (k+2)*x/Q(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 20 2012
G.f. A(x) = G(0) where G(k)= 1 - x*(k+1)/G(k+1); (continued fraction, 1-step).- Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/(1 + x*(Q(0) - 1)/(x+1)) where Q(k)= 1 + (2*k+1)/(1-x/(x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 11 2013
G.f.: Q(0), where Q(k)= 1 + (k+1)*sqrt(x) - sqrt(x)/(1-sqrt(x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
G.f.: Q(0), where Q(k)= 1 + (2*k+1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
a(n) ~ -n^n * 2^(n+1/2) / exp(n). - Vaclav Kotesovec, Feb 23 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
Showing 1-3 of 3 results.

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