A101981 -id:A101981 - cp's OEIS frontend

A002190 Sum_{n>=0} a(n)x^n/n!^2 = -log(BesselJ(0,2sqrt(x))).

0, 1, 1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, 229195817258100, 24730000147369440, 3113066087894608560, 452168671458789789504, 75059305956331837485345, 14121026957032156557396000, 2988687741694684876495689040

Offset: 0

N. J. A. Sloane

Number of non-ambiguous trees, see the Aval et al. reference. - Joerg Arndt, May 11 2015

			-log( Sum_{n>=0} (-x)^n/n!^2 ) = x + x^2/2!^2 + 4*x^3/3!^2 + 33*x^4/4!^2 + 456*x^5/5!^2 + 9460*x^6/6!^2 + ... . -_Paul D. Hanna_, Oct 09 2010

Stany De Smedt, On Sloane's Sequence 1484, Saitama Math. J. 15 (1997), 9-13.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..100
J.-C. Aval, A. Boussicault, M. Bouvel and M. Silimbani, Combinatorics of non-ambiguous trees, 2012. - From _N. J. A. Sloane_, Jan 03 2013
Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, Matteo, Combinatorics of non-ambiguous trees, arXiv:1305.3716 [math.CO], (16-May-2013).
Juan Arias de Reyna, Richard P. Brent and Jan van de Lune, On the sign of the real part of the Riemann zeta-function, arXiv preprint arXiv:1205.4423 [math.NT], 2012.
Beáta Bényi, Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], (2017).
L. Carlitz, A sequence of integers related to the Bessel functions, Proc. Amer. Math. Soc., 14 (1963), 1-9.
William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], 2017.
Mark Dukes, Thomas Selig, Jason P. Smith, Einar Steingrimsson, Permutation graphs and the Abelian sandpile model, tiered trees and non-ambiguous binary trees, arXiv:1810.02437 [math.CO], 2018.
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO], 2006.
Christian Günther, Kai-Uwe Schmidt, Lq norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016.
Joshua F. Robinson and Patrick B. Warren, Renormalisation group theory applied to x^.. + x^. + x^2 = 0, arXiv:2412.16578 [math-ph], 2024. See p. 5.
Index entries for sequences related to Bessel functions or polynomials

Cf. A101981. A diagonal of A217940.

Cf. A115368.

a:= n-> coeff(series(-ln(BesselJ(0,2*sqrt(x))), x, n+1), x, n)*(n!)^2:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 10 2010

nn=18; CoefficientList[Series[-Log[BesselJ[0, 2*Sqrt[x]]], {x, 0, nn}], x]*Table[n!^2, {n, 0, nn}] (* Jean-François Alcover, Jun 22 2011 *)
Clear[q]; q[n_, 1] := (n-1)!^2; q[n_, k_] := q[n, k] = Sum[Binomial[n-1, j]*Binomial[n-1, j+1]*Sum[q[j+1, r]*q[n-j-1, k-r], {r, Max[1, -n+j+k+1], Min[j+1, k-1]}], { n-2}]; a[n_] := q[n, n]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 13 2013 *)

N=66; x='x+O('x^N);
f=-log(sum(n=0,N, (-x)^n/(n!)^2) );
f=serlaplace(f);
f=serlaplace(f);
concat([0],Vec(f))
\\ Joerg Arndt, May 17 2013

\\ Terms starting from a(1)=1:
N=33; B=vector(N);  B[1]=1;  b(j)=B[j+1];
for(n=0,N-2, B[n+2]=sum(i=0,n, my(j=n-i); binomial(n+1,i)*binomial(n+1,j)*b(i)*b(j) ) ); B
\\ Joerg Arndt, May 11 2015

Conjecture: G.f.: 1 = Sum_{n>=0} a(n+1)*A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)^2*(-x)^k. Compare with the following g.f of the Catalan numbers (A000108): 1 = Sum_{n>=0} A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)*(-x)^k. - Paul D. Hanna, Oct 10 2010
a(n) ~ n! * (n-1)! / r^n, where r = 1/4*BesselJZero[0,1]^2 = 1.44579649073669613... - Vaclav Kotesovec, Mar 02 2014
a(0) = 0; a(n) = -(-1)^n + (1/n) * Sum_{k=1..n-1} (-1)^(n-k-1) * binomial(n,k)^2 * k * a(k). - Ilya Gutkovskiy, Jul 15 2021

More terms and better definition from Vladeta Jovovic, Jul 16 2006

Edited by Assoc. Editors of the OEIS, Oct 12 2010

A101980 Matrix logarithm of A008459 (squared entries of Pascal's triangle), read by rows.

Original entry on oeis.org

0, 1, 0, -1, 4, 0, 4, -9, 9, 0, -33, 64, -36, 16, 0, 456, -825, 400, -100, 25, 0, -9460, 16416, -7425, 1600, -225, 36, 0, 274800, -463540, 201096, -40425, 4900, -441, 49, 0, -10643745, 17587200, -7416640, 1430016, -161700, 12544, -784, 64, 0, 530052880, -862143345, 356140800, -66749760, 7239456

Offset: 0

Paul D. Hanna, Dec 23 2004

Column 0 (A101981) is essentially a signed offset version of A002190 and is related to Bessel functions. Row sums form A101982.

			Rows begin:
[0],
[1,0],
[ -1,4,0],
[4,-9,9,0],
[ -33,64,-36,16,0],
[456,-825,400,-100,25,0],
[ -9460,16416,-7425,1600,-225,36,0],
[274800,-463540,201096,-40425,4900,-441,49,0],
[ -10643745,17587200,-7416640,1430016,-161700,12544,-784,64,0],...
and equal the term-by-term product of column 0:
A101981 = {0,1,-1,4,-33,456,-9460,274800,-10643745,...}
with the rows of the squared Pascal's triangle (A008459):
[0],
[1*1^2, 0*1^2],
[ -1*1^2, 1*2^2, 0*1^2],
[4*1^2, -1*3^2, 1*3^2, 0*1^2],
[ -33*1^2, 4*4^2, -1*6^2, 1*4^2, 0*1^2],
[456*1^2, -33*5^2, 4*10^2, -1*10^2, 1*5^2, 0*1^2],...

Cf. A008459, A002190, A101981, A101982.

{T(n,k)=if(nj,binomial(i-1,j-1)^2))^m/m)[n+1,k+1]))}

T(n, k) = A101981(n-k)*C(n, k)^2.

A101982 Row sums of triangle A101980, which is the matrix logarithm of A008459 (squared entries of Pascal's triangle).

Original entry on oeis.org

0, 1, 3, 4, 11, -44, 942, -23561, 806955, -35956868, 2023718198, -140435834681, 11782131588086, -1175694615277233, 137629159046661089, -18679508311308283526, 2909710453923000618155, -515605748075502971981108, 103130355820655917046896638, -23123715029010809457898920545

Offset: 0

Paul D. Hanna, Dec 23 2004

sign

A101981 is essentially a signed offset version of A002190 and is related to Bessel functions.

Cf. A008459, A002190, A101980, A101981.

{a(n)=sum(k=0,n,sum(m=1,n,(-1)^(m-1)* (matrix(n+1,n+1,i,j,if(i>j,binomial(i-1,j-1)^2))^m/m)[n+1,k+1]))}

a(n) = Sum_{k=0..n} A101981(n-k)*C(n, k)^2.

A337824 a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).

Original entry on oeis.org

0, 1, 2, -15, 16, 2505, -60264, -606515, 131316928, -4813100271, -339213768200, 62401665573621, -2075963863814928, -745086903175541927, 140250562903680456332, 808225064553580739325, -5491409141464496462591744, 1013058261721909845376508449, 127689148764914765889971316600

Offset: 0

Ilya Gutkovskiy, Sep 24 2020

sign

Robert Israel, Table of n, a(n) for n = 0..257

Cf. A002190, A033464, A101981, A337590, A337591, A337825.

S:= series(log(1+x*BesselI(0,2*sqrt(x))),x,31):
0,seq(coeff(S,x,n)*(n!)^2, n=1..30); # Robert Israel, Jan 07 2024

a[0] = 0; a[n_] := a[n] = n^2 - (1/n) * Sum[(Binomial[n, k] (n - k))^2 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Log[1 + x BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + x * BesselI(0,2*sqrt(x))).

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n^2 * x^n / (n!)^2).

A337825 a(0) = 0; a(n) = n^3 - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k)^3 * k * a(k).

Original entry on oeis.org

0, 1, 6, -33, -512, 19405, 181116, -45817541, 771776384, 280415588121, -23151651942500, -3217963989270569, 816268626535923936, 38087192839910816485, -43268389662374707851552, 2822720920753640236252875, 3297662826737476255127428096, -833876355494162903256716734927

Offset: 0

Ilya Gutkovskiy, Sep 24 2020

sign

Cf. A002190, A101981, A300452, A337590, A337824, A337826.

a[0] = 0; a[n_] := a[n] = n^3 - (1/n) * Sum[Binomial[n, k]^2 (n - k)^3 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Log[1 + x (BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + x * (BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x)))).

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n^3 * x^n / (n!)^2).

A217567 E.g.f. satisfies: A(x) = ( Sum_{n>=0} x^n/n!^2 )^A(x) where A(x) = Sum_{n>=0} a(n)*x^n/n!^2.

Original entry on oeis.org

1, 1, 5, 73, 2061, 97301, 6897203, 686934284, 91511132653, 15722347919797, 3385861914011775, 893404629519870524, 283510131741909375339, 106536362337513833330932, 46788887175103244923057374, 23747979495191419502491847783, 13795147423164719523469062474093

Offset: 0

Paul D. Hanna, Oct 07 2012

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2!^2 + 73*x^3/3!^2 + 2061*x^4/4!^2 + 97301*x^5/5!^2 +...
where
A(x) = (1 + x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 + x^5/5!^2 +...)^A(x).
Related expansions:
log(A(x)) = x + 3*x^2/2!^2 + 31*x^3/3!^2 + 679*x^4/4!^2 + 25581*x^5/5!^2 + 1474706*x^6/6!^2 + 120670201*x^7/7!^2 + 13298986863*x^8/8!^2 +...
log(A(x))/A(x) = log(1 + x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 +...);
log(A(x))/A(x) = x - x^2/2!^2 + 4*x^3/3!^2 - 33*x^4/4!^2 + 456*x^5/5!^2 - 9460*x^6/6!^2 + 274800*x^7/7!^2 +...+ A101981(n)*x^n/n!^2 +...

Cf. A101981, A179156.

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m/m!^2+x*O(x^n))^A);n!^2*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!^2 satisfies the following formulas.
(1) A(x) = ( Sum_{n>=0} x^n/n!^2 )^A(x).
(2) A(x) = log(A(x)) / ( Sum_{n>=0} A101981(n)*x^n/n!^2 ).

cp's OEIS Frontend

A002190 Sum_{n>=0} a(n)*x^n/n!^2 = -log(BesselJ(0,2*sqrt(x))).

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A101980 Matrix logarithm of A008459 (squared entries of Pascal's triangle), read by rows.

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A101982 Row sums of triangle A101980, which is the matrix logarithm of A008459 (squared entries of Pascal's triangle).

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A337824 a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).

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A337825 a(0) = 0; a(n) = n^3 - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k)^3 * k * a(k).

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A217567 E.g.f. satisfies: A(x) = ( Sum_{n>=0} x^n/n!^2 )^A(x) where A(x) = Sum_{n>=0} a(n)*x^n/n!^2.

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A002190 Sum_{n>=0} a(n)x^n/n!^2 = -log(BesselJ(0,2sqrt(x))).