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.

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A385953 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+4,4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 6, 101, 3756, 271256, 34761512, 7372486163, 2448035959989, 1216747945481685, 872431867857009866, 875060598719254613963, 1196215918953589596769516, 2179513438308809548333358500, 5191611931593198935913809439220, 15896735560092998091331427433546666
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A088716, A351798, A385952, A385954, A385955.
Cf. A385946.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+4, 4)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - Sum_{k=0..4} binomial(4,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385954 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+5,5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 7, 160, 9309, 1193192, 303192604, 140697031749, 111717191583621, 144005113804578040, 288587523313304535136, 867207126292422956078756, 3789698359352103250842742098, 23458242467926487526255374709015, 201037179886862036121457727887328687
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A088716, A351798, A385952, A385953, A385955.
Cf. A385947.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+5, 5)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - Sum_{k=0..5} binomial(5,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385955 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+6,6) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 8, 239, 20595, 4369086, 2027570077, 1877595433603, 3225737601183428, 9693366952072675847, 48534731177400280613882, 388763324236561973987746008, 4812113062706722698140922709260, 89341696197620005494613697916344217, 2424197647354438894347947373843634554628
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A088716, A351798, A385952, A385953, A385954.
Cf. A385948.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+6, 6)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - Sum_{k=0..6} binomial(6,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A112913 Column 2 of triangle A112911.

Original entry on oeis.org

1, 3, 15, 96, 729, 6327, 61380, 655944, 7642557, 96338925, 1306003803, 18947038404, 292956751236, 4810423131828, 83620543389804, 1534493081643552, 29649997606132197, 601828286370137865, 12804746026265284149
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A112911 (triangle), A088716 (column 0), A112912 (column 1), A112914 (column 3).

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+3,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B);return((A^-1)[n+3,3])}

Formula

a(n) = Sum_{k=0..n-1} (n-k+2)*A088716(k)*a(n-k-1) for n>0 with a(0)=1.

A112914 Column 3 of triangle A112911.

Original entry on oeis.org

1, 4, 24, 176, 1492, 14220, 149812, 1724760, 21519108, 289101836, 4160750192, 63873461064, 1042061108096, 18008563271632, 328712410879632, 6320649807989400, 127724091041379492, 2706397280528315148
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A112911 (triangle), A088716 (column 0), A112912 (column 1), A112913 (column 2).

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+4,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B);return((A^-1)[n+4,4])}

Formula

a(n) = Sum_{k=0..n-1} (n-k+3)*A088716(k)*a(n-k-1) for n>0 with a(0)=1.

A158884 G.f. A(x) satisfies: d/dx x*A(x) = 1+x + x*[d/dx log(A(x))].

Original entry on oeis.org

1, 1, -1, 4, -23, 166, -1410, 13602, -145803, 1711690, -21785618, 298370920, -4372151566, 68234087624, -1129894265272, 19788479904366, -365520041466291, 7103187300763530, -144897616964143050, 3096285550330959336
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - x^2 + 4*x^3 - 23*x^4 + 166*x^5 - 1410*x^6 +...
d/dx x*A(x) = 1 + 2*x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 - 9870*x^6 +...
d/dx log(A(x)) = 1 - 3*x + 16*x^2 - 115*x^3 + 996*x^4 - 9870*x^5 +...
Coefficients in powers A(x)^-n begin:
A(x)^-1: (1),-1,2,-7,36,-240,1926,-17815,184916,...;
A(x)^-2: (1),(-2),5,-18,90,-580,4525,-40946,417822,...;
A(x)^-3: 1,(-3),(9),-34,168,-1053,7997,-70776,709614,...;
A(x)^-4: 1,-4,(14),(-56),277,-1700,12594,-109032,1073658,...;
A(x)^-5: 1,-5,20,(-85),(425),-2571,18630,-157860,1526330,...;
A(x)^-6: 1,-6,27,-122,(621),(-3726),26492,-219912,2087658,...;
A(x)^-7: 1,-7,35,-168,875,(-5236),(36652),-298446,2782080,...;
A(x)^-8: 1,-8,44,-224,1198,-7184,(49680),(-397440),3639333,...; ...
where coefficients in parenthesis form A158883 and signed A088716
and A(x)^-1 (first row) is the g.f. of signed A088715.

Links

Crossrefs

Cf. A158883, A088715, A088716.

Programs

  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);Vec(Ser(A)^(n+1)/(n+1))[n+1]}

Formula

G.f. satisfies: x*A'(x) = A(x)*(1+x - A(x))/(A(x) - 1).
G.f.: A(x) = 1/G(-x) where G(x) is the g.f. of A088715.
G.f. satisfies: A(x/F(x)) = F(x) where F(x) is the g.f. of A158883.
G.f. satisfies: A(x*H(-x)) = H(-x) where H(x) is the g.f. of A088716.
G.f. satisfies: [x^n] 1/A(-x)^(n+2) = [x^(n+1)] 1/A(-x)^(n+2)/(n+2) = A088716(n+1).
a(n) ~ -(-1)^n * c * n! * n^2, where c = A238223 / exp(1) = 0.080179614624692622... - Vaclav Kotesovec, Nov 21 2017

A317353 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^3 ).

Original entry on oeis.org

1, 1, 7, 85, 1391, 27852, 650161, 17204220, 506849175, 16418826823, 579491009098, 22126506842782, 908765624923517, 39955891685644210, 1872870160239756648, 93252093750481971120, 4916210973540632697015, 273627175793531458018185, 16036039197194394497206285, 987180880355901292395754555, 63693966375385851829434069094
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 7*x^2 + 85*x^3 + 1391*x^4 + 27852*x^5 + 650161*x^6 + 17204220*x^7 + 506849175*x^8 + 16418826823*x^9 + ...
where A(x) = 1 + x*A(x)^4 + 3*x^2*A(x)^3*A'(x).
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*x*A(x)^3 ) / A(x) begins:
n=1: [1, 0, -7, -326, -23619, -2514244, -367699955, -70373452722, ...];
n=2: [1, 1, 0, -184, -17344, -2038008, -313614848, -61905013280, ...];
n=3: [1, 2, 9, 0, -9555, -1469682, -250952499, -52305865308, ...];
n=4: [1, 3, 20, 232, 0, -795496, -178628288, -41455820640, ...];
n=5: [1, 4, 33, 518, 11597, 0, -95428955, -29221983590, ...];
n=6: [1, 5, 48, 864, 25536, 934056, 0, -15457466592, ...];
n=7: [1, 6, 65, 1276, 42141, 2025842, 109167637, 0, ...];
n=8: [1, 7, 84, 1760, 61760, 3296568, 233751616, 17329568992, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Links

Crossrefs

Cf. A088716, A317352, A317354.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = 1 + x*A*(x*A^3)' +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)^3 ) * (2 - 1/Ser(A)))[#A] );H=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)^3 ) * (m+1 - m/Ser(A)))[#A]/m );H=A; 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)^3 ).
(2) [x^n] exp( n * x*A(x)^3 ) / A(x) = 0 for n>0.
(3.a) [x^n] exp(-n * x*A(x)^3) * (2 - 1/A(x)) = 0 for n >= 1.
(3.b) [x^n] exp(-n^2 * x*A(x)^3) * (n + 1 - n/A(x)) = 0 for n >= 1.
(3.c) [x^n] exp(-n^(p+1) * x*A(x)^3) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.
a(n) ~ c * 3^n * n! * n^(4/3), where c = 0.147653317889315463510794506... - Vaclav Kotesovec, Aug 09 2018

A317354 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^4 ).

Original entry on oeis.org

1, 1, 9, 143, 3077, 81186, 2500555, 87388600, 3402833181, 145793616851, 6809846205490, 344295007127226, 18732966469820743, 1091592281265899216, 67839235267146659208, 4479984759698760684608, 313349016085369145384269, 23145049153829729002326633, 1800552901345736894764054891, 147168026141415485071183102785
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2018

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 9*x^2 + 143*x^3 + 3077*x^4 + 81186*x^5 + 2500555*x^6 + 87388600*x^7 + 3402833181*x^8 + ...
where A(x) = 1 + x*A(x)^5 + 4*x^2*A(x)^4*A'(x).
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*x*A(x)^4 ) / A(x) begins:
n=1: [1, 0, -9, -554, -52995, -7440324, -1434977405, ...]
n=2: [1, 1, 0, -304, -38032, -5920008, -1205232128, ...];
n=3: [1, 2, 11, 0, -20475, -4189122, -949414689, ...];
n=4: [1, 3, 24, 364, 0, -2224296, -665082368, ...];
n=5: [1, 4, 39, 794, 23741, 0, -349575245, ...];
n=6: [1, 5, 56, 1296, 51120, 2511576, 0, ...];
n=7: [1, 6, 75, 1876, 82533, 5340642, 386787007, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Links

Crossrefs

Cf. A088716, A317352, A317353.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = 1 + x*A*(x*A^4)' +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)^4 ) * (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)^4 ) * (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)^4 ).
(2) [x^n] exp( n * x*A(x)^4 ) / A(x) = 0 for n>0.
(3.a) [x^n] exp(-n * x*A(x)^4) * (2 - 1/A(x)) = 0 for n >= 1.
(3.b) [x^n] exp(-n^2 * x*A(x)^4) * (n + 1 - n/A(x)) = 0 for n >= 1.
(3.c) [x^n] exp(-n^(p+1) * x*A(x)^4) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.
a(n) ~ c * 4^n * n! * n^(5/4), where c = 0.1196141591155152938065262... - Vaclav Kotesovec, Aug 09 2018

A376096 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 9, 260, 17215, 2189997, 477731884, 164858203944, 84745577983095, 61951785517193675, 62077057930391945969, 82749694746019635920952, 143157935882304543684640676, 314805573970543375502985796300, 864458294787075036217714712292600, 2919280453922974335841433174057739408
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2024

Keywords

Crossrefs

Cf. A000699, A015084, A088716, A256019, A376095, A376097.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 7 x^2 A[x] A'[x] + 6 x^3 A[x] A''[x] + x^4 A[x] A'''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 7 * x^2 * A(x) * A'(x) + 6 * x^3 * A(x) * A''(x) + x^4 * A(x) * A'''(x).

A193332 E.g.f. satisfies: A(x) = x*exp( A(x)/A'(x) ).

Original entry on oeis.org

1, 2, -3, 52, -1315, 50286, -2655863, 183322952, -15928677063, 1695597280570, -216636191518219, 32688113040335292, -5749136647259226923, 1165789270581830003942, -270019628802455686919295, 70862777375461690495134736, -20921819854506620454336189583
Offset: 1

Views

Author

Paul D. Hanna, Jul 23 2011

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! - 3*x^3/3! + 52*x^4/4! - 1315*x^5/5! + 50286*x^6/6! - 2655863*x^7/7! + 183322952*x^8/8! +...
where A(x)/A'(x) = log(A(x)/x) equals the integer series:
(1) A(x)/A'(x) = x - x^2 + 3*x^3 - 14*x^4 + 85*x^5 - 621*x^6 + 5236*x^7 - 49680*x^8 + 521721*x^9 - 5994155*x^10 +...
which equals -G(-x) where G(x) is the g.f. of A088716.
The series reversion, -L(-x), begins:
(2) -L(-x) = x - 2*x^2/2! + 15*x^3/3! - 232*x^4/4! + 5905*x^5/5! - 220176*x^6/6! + 11210479*x^7/7! - 743759360*x^8/8! +...
where L(x) is the e.g.f. A161968.

Links

Crossrefs

Cf. A088716, A161968, A161967.

Programs

  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=x*exp(A/(A'+x*O(x^n))));n!*polcoeff(A,n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x)/A'(x) = -G(-x) where G(x) = x + x*G(x)*G'(x) is a g.f. of A088716; thus, log(A(x)/x) is an integer series.
(2) A(-L(-x)) = x where L(x) = x*exp(x*L'(x)) is the e.g.f. of A161968.
a(n) ~ c * (-1)^n * (n!)^2, where c = 0.217950789447151065... (see A238223). - Vaclav Kotesovec, Feb 26 2014
Previous Showing 31-40 of 55 results. Next

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