A112912 -id:A112912 - cp's OEIS frontend

A088716 G.f. satisfies: A(x) = 1 + xA(x)d/dx[xA(x)] = 1 + xA(x)^2 + x^2A(x)A'(x).

1, 1, 3, 14, 85, 621, 5236, 49680, 521721, 5994155, 74701055, 1003125282, 14437634276, 221727608284, 3619710743580, 62605324014816, 1143782167355649, 22014467470369143, 445296254367273457, 9444925598142843970

Offset: 0

Paul D. Hanna, Oct 12 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
M. J. H. Al-Kaabi, Monomial Bases for free pre-Lie algebras, Sem. Lothar. Comb. 71 (2014) B71b.
Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 13.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 11.

Cf. A112916 (A^2), A112911, A112912, A112913, A112914.
Cf. A088715, A182962, A112915, A218222, A238223.
Cf. A300736, A300987, A300989, A300991, A300993.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(j)*a(n-j-1)*(j+1), j=0..n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 10 2017

a=ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = Sum[k*a[[n-k+1]]*a[[k]],{k,1,n}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 21 2014 *)
m = 20; A[_] = 0;
Do[A[x_] = 1 + x A[x]^2 + x^2 A[x] A'[x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *)
a[1]:=1; a[2]:=1; a[n_]:=a[n]=n/2 Sum[a[k] a[n-k], {k,1,n-1}];
Map[a,Range[20]] (* Oliver Seipel, Nov 03 2024 ,after Schröder 1870 *)

a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*a(k)*a(n-k-1)))

{a(n)=local(G=1+x);for(i=1,n,G=exp(x/(1 - x*deriv(G)/G+x*O(x^n))));polcoeff(log(G)/x,n)} \\ Paul D. Hanna, Jan 01 2011

a(n) = Sum_{k=1..n} k*a(k-1)*a(n-k) for n>=1 with a(0)=1.
Forms column 0 of triangle T=A112911, where the matrix inverse satisfies [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0.
Self-convolution is A112916, where a(n) = (n+1)/2*A112916(n-1) for n>0.
G.f.: A(x) = serreverse(x/f(x))/x where f(x) is the g.f. of A088715.
O.g.f.: A(x) = log(G(x))/x where G(x) is the e.g.f. of A182962 given by G(x) = exp( x/(1 - x*G'(x)/G(x)) ). [Paul D. Hanna, Jan 01 2011]
O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) / A(x) = 0 for n>0. - Paul D. Hanna, May 25 2018
O.g.f. A(x) satisfies [x^n] exp( n * x*A(x) ) * (1 - n*x) = 0 for n>0. - Paul D. Hanna, Jul 24 2019
From Paul D. Hanna, Jul 20 2018 (Start):
O.g.f. A(x) satisfies:
* [x^n] exp(-n * x*A(x)) * (2 - 1/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^2 * x*A(x)) * (n + 1 - n/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^(p+1) * x*A(x)) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0. (End)
a(n) ~ c * n! * n^2, where c = 0.21795078944715106549282282244231982088... (see A238223). - Vaclav Kotesovec, Feb 21 2014

A112911 Triangle T, read by rows, such that the matrix inverse satisfies: [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0, with T(n,n)=1 for n>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 85, 44, 15, 4, 1, 621, 298, 96, 24, 5, 1, 5236, 2358, 729, 176, 35, 6, 1, 49680, 21154, 6327, 1492, 290, 48, 7, 1, 521721, 211100, 61380, 14220, 2725, 444, 63, 8, 1, 5994155, 2313030, 655944, 149812, 28425, 4590, 644, 80, 9, 1

Offset: 0

Paul D. Hanna, Oct 06 2005

Matrix inverse square satisfies: [T^-2](3*n+2,n) = 0 for n>=0.

			Triangle T begins:
1;
1,1;
3,2,1;
14,8,3,1;
85,44,15,4,1;
621,298,96,24,5,1;
5236,2358,729,176,35,6,1;
49680,21154,6327,1492,290,48,7,1; ...
Matrix inverse T^-1 begins:
1;
-1,1;
-1,-2*1,1;
-3,-2*1,-3*1,1;
-14,-2*3,-3*1,-4*1,1;
-85,-2*14,-3*3,-4*1,-5*1,1;
-621,-2*85,-3*14,-4*3,-5*1,-6*1,1; ...
where [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0.

Cf. A088716 (column 0), A112912 (column 1), A112913 (column 2), A112914 (column 3).

{T(n,k)=local(A=Mat(1),B); for(m=2,n+1,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+1,k+1])}

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

Paul D. Hanna, Oct 06 2005

nonn

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

{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])}

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

Paul D. Hanna, Oct 06 2005

nonn

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

{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])}

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

A122913 Minimum numbers k such that (k^2*2^n + 1) is prime.

Original entry on oeis.org

1, 1, 3, 1, 6, 2, 3, 1, 6, 5, 3, 4, 12, 2, 6, 1, 3, 10, 15, 5, 9, 5, 18, 25, 9, 13, 9, 14, 12, 7, 6, 9, 3, 17, 9, 9, 15, 12, 9, 6, 6, 3, 3, 11, 42, 18, 21, 9, 66, 10, 33, 5, 27, 7, 48, 80, 24, 40, 12, 20, 6, 10, 3, 5, 3, 7, 3, 79, 75, 63, 96, 40, 48, 20, 24, 10, 12, 5, 6, 15, 3, 22, 72, 11

Offset: 1

Alexander Adamchuk, Sep 18 2006

nonn

3 divides a(2k+1) for k>0. Corresponding primes of the form (k^2*2^n + 1) are listed in A122912[n] = {3,5,73,17,1153,257,1153,257,18433,25601,18433,65537,1179649,65537,1179649,65537,1179649,26214401,117964801,...}. There are repeating patterns in a(n) such that for many n a(n) = 2*a(n+2) and a(n+1) = 2*a(n+3). For example, {6,2,3,1}, {12,2,6,1}, {42,18,21,9}, {96,40,48,20,24,10,12,5,6}, {66,10,33,5}, {48,80,24,40,12,20,6,10,3}, {366,38,183,19}. These patterns correspond to identical twin runs in A122912[n] such that A122912[n] = A122912[n+2] and A122912[n+1] = A122912[n+3]. The final index of many such twin runs is perfect power such as {8,16,25,64,81,100,...}.

Cf. A112912.

mnk[n_]:=Module[{k=1,c=2^n},While[!PrimeQ[k^2 c+1],k++];k]; Array[mnk,90] (* Harvey P. Dale, Jul 22 2025 *)

a(n) = Sqrt[ (A122912[n] - 1) / 2^n ].

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A088716 G.f. satisfies: A(x) = 1 + x*A(x)*d/dx[x*A(x)] = 1 + x*A(x)^2 + x^2*A(x)*A'(x).

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A112911 Triangle T, read by rows, such that the matrix inverse satisfies: [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0, with T(n,n)=1 for n>=0.

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A112913 Column 2 of triangle A112911.

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A112914 Column 3 of triangle A112911.

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A122913 Minimum numbers k such that (k^2*2^n + 1) is prime.

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Formula

A088716 G.f. satisfies: A(x) = 1 + xA(x)d/dx[xA(x)] = 1 + xA(x)^2 + x^2A(x)A'(x).