A291843 -id:A291843 - cp's OEIS frontend

A294166 Row sums of A291843.

1, 0, 1, 8, 71, 789, 10365, 157031, 2692497, 51519756, 1088093185, 25140587651, 630820490833, 17082650998878, 496596665961713, 15425333714935513, 509890407550644949, 17871584701588777344, 662057571007292023593, 25847670560115633381442

Offset: 0

Gheorghe Coserea, Nov 05 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..303

Cf. A291843.

A291843_ser(N, t='t) = {
  my(x='x+O('x^N), y=1, y1=0, n=1,
  dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
  while (n++,
   y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
        (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
   if (y1 == y, break); y = y1; ); y;
};
Vec(A291843_ser(20,1))

G.f. y(x) satisfies: 0 = 2*x^2*(1+x)*y*deriv(y,x) + x*y^2 - (1+x)^2*(1-2*x)*y + (1+x)*(1-2*x).

A294167 Column 1 of triangle A291843.

Original entry on oeis.org

3, 33, 388, 5101, 75444, 1248911, 22964112, 465344235, 10316541393, 248583207948, 6472094085480, 181133509590584, 5424172954377851, 173089061380034193, 5864328868997378224, 210259284591708083349, 7954221480382049449284, 316654854011156144727459, 13233287747652092826502116

Offset: 3

Gheorghe Coserea, Nov 06 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 3..305

Cf. A291843.

A291843_ser(N, t='t) = {
  my(x='x+O('x^N), y=1, y1=0, n=1,
  dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
  while (n++,
   y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
        (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
   if (y1 == y, break); y = y1; ); y;
};
A291843_kol(k, N=19) = {
  my(s = A291843_ser(N+1+3*(k+1)\2, t='t + O('t^(k+1))));
  Ser(polcoeff(s, k, 't), 'x, N);
};
Vec(A291843_kol(1))

A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.

Original entry on oeis.org

1, 2, 20, 352, 8624, 266784, 9896448, 426577920, 20918138624, 1149216540160, 69911382901760, 4665553152081920, 338942971881472000, 26631920159494995968, 2250690001888540950528, 203595258621775065120768, 19629810220331494121865216

Offset: 0

Noam Zeilberger, Jan 21 2016

nonn

A linear lambda term is indecomposable if it has no closed proper subterm.
Equivalently, number of closed bridgeless rooted trivalent maps (on compact oriented surfaces of arbitrary genus) with 2n+1 trivalent vertices (and 1 univalent vertex).
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

			A(x) = 1 + 2*x + 20*x^2 + 352*x^3 + 8624*x^4 + 266784*x^5 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..303
Lawrence Dresner, Protection of a test magnet wound with a Ag/BSCCO high-temperature superconductor, Oak Ridge National Lab technical report (ORNL/HTSPC-3), 1992. See Eq. (25).
Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv:1512.06751 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. For the video see http://noamz.org/videos/expmath.2020.06.18.mp4.
Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. [Local copy]

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.

Cf. A000309, A062980.

a[0] = 1; a[1] = 2; a[n_] := a[n] = (6n-2) a[n-1] + Sum[(6k+2) a[k] a[n-1-k], {k, 1, n-2}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 16 2018, after Gheorghe Coserea *)

seq(N) = {
  my(a = vector(N)); a[1] = 2;
  for(n=2, N,
    a[n] = (6*n-2)*a[n-1] + sum(k=1, n-2, (6*k+2)*a[k]*a[n-1-k]));
  concat(1,a);
};
seq(16)
\\ test 1: y = x^2*subst(Ser(seq(201)),'x,-'x^6); 0 == x^5*y*y' + y - x^2
\\ test 2: y = Ser(seq(201)); 0 == 6*y*y'*x^2 + 2*y^2*x - y + 1
\\ Gheorghe Coserea, Nov 10 2017
F(N) = {
  my(x='x+O('x^N), t='t, F0=x, F1=0, n=1);
  while(n++,
    F1 = t + x*(F0 - subst(F0,t,0))^2 + x*deriv(F0,t);
    if (F1 == F0, break()); F0 = F1;);
  F0;
};
seq(N) = my(v=Vec(subst(F(2*N+2),'t,0))); vector((#v+1)\2, n, v[2*n-1]);
seq(16) \\ Gheorghe Coserea, Apr 01 2017

The o.g.f. f(z) = z + 2*z^3 + 20*z^5 + 352*z^7 + ... can be defined using a catalytic variable as f(z) = F(z,0), where F(z,x) satisfies the functional-differential equation F(z,x) = x + z*(F(z,x) - F(z,0))^2 + z*(d/dx)F(z,x).
From Gheorghe Coserea, Nov 10 2017: (Start)
0 = x^5*y*y' + y - x^2, where y(x) = x^2*A(-x^6).
0 = 6*y*y'*x^2 + 2*y^2*x - y + 1, where y(x) = A(x).
a(n) = (6*n-2)*a(n-1) + Sum_{k=1..n-2} (6*k+2)*a(k)*a(n-1-k), for n >= 2.
(End)
a(n) = A291843(3*n+1, 2*n), n >= 1. - Danny Rorabaugh, Nov 10 2017

A291844 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 4, 2, 29, 23, 274, 292, 36, 3145, 4068, 994, 16, 42294, 62861, 22250, 1512, 651227, 1075562, 484840, 61027, 1060, 11295242, 20275944, 10867381, 1977879, 93188, 280, 217954807, 418724047, 255929070, 59896915, 4823178, 80632, 4632600152, 9418874022, 6387031115, 1798212190, 204846125, 7410676, 37056, 107572674851, 229535650138, 169414005231, 55017177704, 8022471066, 463514918, 7255380, 7040

Offset: 0

Gheorghe Coserea, Oct 24 2017

Row n>0 contains floor((2*n+2)/3) terms.

			A(x;t) = 1 + x + (4 + 2*t)*x^2 + (29 + 23*t)*x^3 + (274 + 292*t + 36*t^2)*x^4 + ...
Triangle starts:
n\k  [0]        [1]        [2]        [3]       [4]      [5]
[0]  1;
[1]  1;
[2]  4,         2;
[3]  29,        23;
[4]  274,       292,       36;
[5]  3145,      4068,      994,       16;
[6]  42294,     62861,     22250,     1512;
[7]  651227,    1075562,   484840,    61027,    1060;
[8]  11295242,  20275944,  10867381,  1977879,  93188,   280;
[9]  217954807, 418724047, 255929070, 59896915, 4823178, 80632;
[10] ...

Gheorghe Coserea, Rows n = 0..124, flattened
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A286795, A286798, A286800, A291843.

Columns k=0..5 give A294160 (k=0), A294161 (k=1), A294162 (k=2), A294163 (k=3), A294164 (k=4), A294165 (k=5).

m = maxExponent = 13; Z[_] = 0;
Do[Z[t_] = -(((1 - l + l (1+t) Z[t]) (-((t Z[t])/(1 + l t)) - (1 - t - 2 l t^2)/(1 - l + l (1+t) Z[t]) - 2 t^2 Z'[t]))/((1+t) (1 - t - 2 l t^2))) + O[t]^m // Normal // Simplify, {m}];
gamma[t_] = ((1 + l t)(-1 + Z[t] + t Z[t]))/(Z[t]^2 (t + l t (-1 + Z[t] + t Z[t]))) + O[t]^m // Normal // Simplify;
CoefficientList[# + O[l]^m, l]& /@ Most @ CoefficientList[gamma[t], t] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

A291843_ser(N, t='t) = {
  my(x='x+O('x^N), y=1, y1=0, n=1,
  dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
  while (n++,
   y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
        (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
   if (y1 == y, break); y = y1;); y;
};
A291844_ser(N, t='t) = {
  my(z = A291843_ser(N+1,t));
  ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
concat(apply(p->Vecrev(p), Vec(A291844_ser(12))))

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y = ((1+x)*z - 1) * (1 + t*x)/((1-t + t*(1+x)*z)*x*z^2), where z = A291843(x;t) and P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n > 0.

A294158(n) = P_n(1), A294159(n)=P_n(-1), A294160(n)=P_n(0).

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A294166 Row sums of A291843.

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A294167 Column 1 of triangle A291843.

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A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.

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A291844 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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