A002190 -id:A002190 - cp's OEIS frontend

A115368 Decimal expansion of first zero of the Bessel function J_0(z).

2, 4, 0, 4, 8, 2, 5, 5, 5, 7, 6, 9, 5, 7, 7, 2, 7, 6, 8, 6, 2, 1, 6, 3, 1, 8, 7, 9, 3, 2, 6, 4, 5, 4, 6, 4, 3, 1, 2, 4, 2, 4, 4, 9, 0, 9, 1, 4, 5, 9, 6, 7, 1, 3, 5, 7, 0, 6, 9, 9, 9, 0, 9, 0, 5, 9, 6, 7, 6, 5, 8, 3, 8, 6, 7, 7, 1, 9, 4, 0, 2, 9, 2, 0, 4, 4, 3, 6, 3, 4, 3, 7, 6, 0, 1, 4, 5, 2, 5, 4, 7, 8, 6, 8, 9

Offset: 1

Eric W. Weisstein, Jan 21 2006

"This [constant] arises from the study of a vibrating, homogeneous membrane that is uniformly stretched across the unit disk. [Its square] is the principal frequency of the sound one hears when a kettledrum is struck." - Quoted from the book by Steven R. Finch.

Siegel proves (the Main Theorem) that J_0(z) is transcendental if z is algebraic and nonzero, but since in our case J_0(z) = 0 is not transcendental it follows that z cannot be algebraic. - Charles R Greathouse IV, Oct 20 2020

			2.4048255576957727686...

Chi Keung Cheung et al., Getting Started with Mathematica, 2nd Ed. New York: J. Wiley (2005) p. 7.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 221.
C. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929/30, No. 1. Translated as "On some applications
of Diophantine approximations" by Clemens Fuchs.

Eric Weisstein's World of Mathematics, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115369, A115370, A115371, A115372, A115373.

Cf. A000275, A002190, A188489, A181167, A181168.

RealDigits[BesselJZero[0, 1], 10, 120][[1]] (* Alonso del Arte, May 06 2011 *)

solve(x=2,3,besselj(0,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(0) \\ Charles R Greathouse IV, Aug 06 2022

A101981 Column 0 of triangle A101980, which is the matrix logarithm of A008459 (squared entries of Pascal's triangle).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Dec 23 2004

sign

This sequence is a signed version of A002190 and is related to Bessel functions.

Alois P. Heinz, Table of n, a(n) for n = 0..100
O. Arizmendi, T. Hasebe, F. Lehner, C. Vargas, Relations between cumulants in noncommutative probability, arXiv preprint arXiv:1408.2977 [math.CO], 2014.
Christian Günther, Kai-Uwe Schmidt, Lq norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016.

Cf. A008459, A002190, A101980, A101982.

a:= n-> (-1)^(n+1)*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 27 2012

a[n_] := (-1)^(n+1) n!^2 SeriesCoefficient[-Log[BesselJ[0, 2 Sqrt[x]]], {x, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 26 2018 *)

{a(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+1,1])}

a(n) = (-1)^(n+1)*A002190(n) for n>=0.

a(n) = 1 - Sum_{j=1..k-1} binomial(k, j)*binomial(k-1, j-1)*a(j) for n >= 1. See Günther & Schmidt link p.5. - Michel Marcus, Jun 17 2017

A238390 E.g.f.: x / BesselJ(1, 2*x) (even powers only).

Original entry on oeis.org

1, 1, 4, 35, 546, 13482, 485892, 24108513, 1576676530, 131451399794, 13609184032808, 1712978776719938, 257612765775847132, 45620136452519144700, 9396239458048330569840, 2227147531572856811691105, 601916577165056911293330930, 183994483721828524163677628370

Offset: 0

Vaclav Kotesovec, Mar 01 2014

nonn

Aerated, the e.g.f. is e^(a.t) =  1/AC(i*t) = 1/[I_1(2i*t)/(it)] = 1/Sum_{n>=0} (-1)^n t^(2n) / [n!(n+1)!] = a_0 + a_2 t^2/2! + a_4 t^4/4! + ... = 1 + t^2/2! + 4 t^4/4! + 35 t^6/6! + ..., where AC(t) is the e.g.f. for the aerated Catalan numbers c_n of A126120 and I_n(t) are the modified Bessel functions of the first kind (i = sqrt(-1)). The signed, aerated sequence b_n = (i)^n a_n has the e.g.f. e^(b.t) = 1/AC(t) and, therefore, (i*a. + c.)^n = Sum_{k=0..n} binomial(n,k) i^k a_k c_(n-k) vanishes except for n=0 for which it's unity. - Tom Copeland, Jan 23 2016
With q(n) = A126120(n+1) and q(0) = 0, d(2n) =  (-1)^n A238390(n) and zero for odd arguments, and r(2n+1) = (-1)^n A180874(n+1) and zero for even arguments, then r(n) = (q. + d.)^n = Sum_{k=0..n} binomial(n,k) q(k) d(n-k), relating these sequences (and A000108) through binomial convolutions. Then also, (r. + c. + d.)^n = r(n). See A180874 for proofs and for relations to A097610. For quick reference, q = (0, 1, 0, 2, 0, 5, 0, 14, ..), d = (1, 0, -1, 0, 4, 0, -35, 0, ..), and r = (0, 1, 0, -1, 0, 5, 0, -56, ..). - Tom Copeland, Jan 28 2016
Aerated and signed, this sequence contains the moments m(n) of the Appell polynomial sequence UMT(n,h1,h2) that is the umbral compositional inverse of the Appell sequence of Motzkin polynomials MT(n,h1,h2) of A097610 with exp[x UMT(.,h1,h2)] = e^(x*h1) / AC(x*y) where y = sqrt(h2) and AC is defined above. UMT(n,h1,h2) = (m.y + h1)^n with (m.)^(2n) = m(2n) = (-1)^n A238390(n) and zero otherwise. Consequently, the associated lower triangular matrices A007318(n,k)*m(n-k) and A007318(n,k)*A126120(n-k) form an inverse pair (cf. also A133314), and MT(n,UMT(.,h1,h2),h2) = h1^n = UMT(n,MT(.,h1,h2),h2). - Tom Copeland, Jan 30 2016

G. C. Greubel, Table of n, a(n) for n = 0..98

Cf. A103365, A180874, A115369, A000275, A002190.

Cf. A000108, A007318, A097610, A126120, A133314.

S:= series(x/BesselJ(1,2*x),x,102):
seq((2*j)!*coeff(S,x,2*j),j=0..50); # Robert Israel, Jan 31 2016

Table[(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 40}], x] * Range[0, 40]!)[[n]], {n, 1, 41, 2}]

a(n) ~ c * (n!)^2 / (sqrt(n) * r^n), where r = BesselJZero[1, 1]^2/16 = 0.91762316513274332857623611, and c = 1/(Sqrt[Pi]*BesselJ[2, BesselJZero[1, 1]]) = 1.4008104828035425937394082168... - Vaclav Kotesovec, Mar 01 2014, updated Apr 01 2018

A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

Original entry on oeis.org

1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236, 782369683393860737, 6355271576489378132, 51624542295308885793

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,1).

Cf. A002018, A002190, A013192, A028576, A041024, A041027.

Cf. A053120, A058153, A058155, A072754, A075132.

[n le 2 select 4^(n-1) else 8*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022

LinearRecurrence[{8,1},{1,4},30] (* or *) With[{c=Sqrt[17]},Simplify/@ Table[1/2 (c-4)((c+4)^n-(4-c)^n (33+8c)),{n,30}]] (* Harvey P. Dale, May 07 2012 *)

a[0]:1$ a[1]:4$ a[n]:=8*a[n-1]+a[n-2]$ A088317(n):=a[n]$
makelist(A088317(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

A088317=BinaryRecurrenceSequence(8,1,1,4)
[A088317(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

a(n) = ( (4+sqrt(17))^n + (4-sqrt(17))^n )/2.
a(n) = A086594(n)/2.
Lim_{n -> oo} a(n+1)/a(n) = 4 + sqrt(17).
From Paul Barry, Nov 15 2003: (Start)
E.g.f.: exp(4*x)*cosh(sqrt(17)*x).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k).
a(n) = (-i)^n * T(n, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
a(n) = A041024(n-1), n>0. - R. J. Mathar, Sep 11 2008
G.f.: (1-4*x)/(1-8*x-x^2). - Philippe Deléham, Nov 16 2008 and Nov 20 2008
a(n) = (1/2)*((33+8*sqrt(17))*(4-sqrt(17))^(n+2) + (33-8*sqrt(17))*(4+sqrt(17))^(n+2)). - Harvey P. Dale, May 07 2012

A181167 G.f.: 1 = Sum_{n>=0} a(n)x^n Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

Original entry on oeis.org

1, 1, 8, 165, 6384, 397320, 36273600, 4566166605, 757975618400, 160424015864112, 42164387189608320, 13473505313334666600, 5144136790654611953280, 2312696796696904699224000, 1209297981696245764641077760, 727688337054213932985609546525

Offset: 0

Paul D. Hanna, Oct 08 2010

nonn

Compare g.f. to a g.f of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n * Sum_{k>=0} C(2n+k,k)*(-x)^k.

			E.g.f.: E(x) = 1 + x^2/2! + 8*x^4/4! + 165*x^6/6! + 6384*x^8/8! +...
where the e.g.f. equals the continued fraction:
E(x) = 1/(1 - x^2/(2 - x^2/(3 - x^2/(4 - x^2/(5 - x^2/(6 - x^2/(7 - x^2/(8 - x^2/(9 - x^2/(10 - ...)))))))))). [Due to Matthieu Josuat-Vergès]
Illustrate the g.f. by the series:
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 - 21^2*x^5 +...)
+ 8*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 - 126^2*x^5 +...)
+ 165*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 - 462^2*x^5+...)
+ 6384*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 397320*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 36273600*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 4566166605*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4 +...)
+...
Compare to a g.f. of the Catalan numbers (A000108):
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*x*(1 - 3*x + 6*x^2 - 10*x^3 + 15*x^4 - 21*x^5 +...)
+ 2*x^2*(1 - 5*x + 15*x^2 - 35*x^3 + 70*x^4 - 126*x^5 +...)
+ 5*x^3*(1 - 7*x + 28*x^2 - 84*x^3 + 210*x^4 - 462*x^5 +...)
+ 14*x^4*(1 - 9*x + 45*x^2 - 165*x^3 + 495*x^4 - 1287*x^5 +...)
+ 42*x^5*(1 - 11*x + 66*x^2 - 286*x^3 + 1001*x^4 - 3003*x^5 +...)
+ 132*x^6*(1 - 13*x + 91*x^2 - 455*x^3 + 1820*x^4 - 6188*x^5 +...)
+...
Surprisingly, terms a(n) are divisible by n*A000108(n) for n>0:
a(2)=2*2*2, a(3)=3*5*11, a(4)=4*14*114, a(5)=5*42*1892, a(6)=6*132*45800, a(7)=7*429*1520535, ..., a(n)=n*A000108(n)*A181168(n).

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Cf. A181168, A180716.
Cf. A002190. [From Paul D. Hanna, Oct 09 2010]
Cf. A115368.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, x/(y+1)*(b(x-1, y-1)+b(x-1, y+1))))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 08 2018

nmax=20; Table[(CoefficientList[Series[BesselJ[1,2*x]/x/BesselJ[0,2*x],{x,0,2*nmax}],x] * Range[0,2*nmax]!)[[2*n-1]],{n,1,nmax}] (* Vaclav Kotesovec, Jul 31 2014 *)

{a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, n-m, binomial(2*m+k, k)^2*(-x)^k)+x*O(x^n)), n))}

/* Formula: a(n) = A000108(n)*A002190(n+1) implies: */
{a(n)=binomial(2*n,n)/(n+1)*(n+1)!^2*polcoeff(-log(sum(m=0,n+1,(-x)^m/m!^2)+O(x^(n+2))),n+1)} \\ Paul D. Hanna, Oct 09 2010

/* Continued Fraction expansion of the E.G.F.: */
{a(n)=local(CF=1+O(x));for(i=0,n,CF=1/((n-i+1)-x^2*CF));(2*n)!*polcoeff(CF,2*n)}

a(n) = n*A000108(n)*A181168(n) = C(2n,n-1)*A181168(n) for n>0, with a(0)=1.
a(n) = A000108(n)*A002190(n+1), where A002190 describes the coefficients in -log(BesselJ(0,2*sqrt(x))) and A000108 is the Catalan numbers. - Paul D. Hanna, Oct 09 2010
Differentiating -log(BesselJ(0,2*sqrt(x))) and substituting z=z^2 gives the e.g.f. Sum_{n>=0} a(n) * z^(2*n)/(2n)! = BesselJ(1,2*z)/z/BesselJ(0,2*z). Consequently, using Gauss' continued fraction, this e.g.f. is also: 1/(1-z^2/(2-z^2/(3-z^2/(4-z^2/(5-z^2/...))))). - Matthieu Josuat-Vergès, Apr 17 2011
E.g.f.: U(0) where U(k) = 1 - x^2/(x^2 - (k+1)*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 15 2012
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 16/BesselJZero[0,1]^2 = 2.76664110449031883070186935..., c = 4/(sqrt(Pi)*BesselJZero[0,1]^2) = 0.390227523142124366836071453... . - Vaclav Kotesovec, Jul 31 2014

A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

Original entry on oeis.org

1, 1, 4, 1, 18, 36, 1, 68, 432, 576, 1, 250, 3900, 14400, 14400, 1, 922, 32400, 252000, 648000, 518400, 1, 3430, 262542, 3880800, 19404000, 38102400, 25401600, 1, 12868, 2119152, 56664384, 493920000, 1795046400, 2844979200, 1625702400

Offset: 1

Peter Bala, Jul 11 2011

Compare with triangle A019538, whose entries are given by
... Sum multinomial(n; n_1,n_2,...,n_k), where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.
For related tables see A061691 and A192721.
Let P be the poset of all ordered pairs (S,T) of subsets of [n] with |S|=|T|, ordered componentwise by inclusion.  T(n,k) is the number of length k chains in P from ({},{}) to ([n],[n]). - Geoffrey Critzer, Apr 15 2020

			The triangle begins
n/k|..1.....2.......3........4........5........6
================================================
.1.|..1
.2.|..1.....4
.3.|..1....18.....36
.4.|..1....68.....432......576
.5.|..1...250....3900....14400....14400
.6.|..1...922...32400...252000...648000...518400
...
T(4,2) = 68:
There are 3 compositions of 4 into 2 parts, namely, 4 = 2 + 2 = 1 + 3 = 3 + 1; hence
T(4,2) = (4!/(2!*2!))^2 + (4!/(1!*3!))^2 + (4!/(3!*1!))^2
= 36 + 16 + 16 = 68.
Matrix identity: A192721 * Pascal's triangle = row reverse of A192722:
/...1................\ /..1..............\
|...3.....1...........||..1....1..........|
|..19....16.....5.....||..1....2....1.....|
|.211...299....65....1||..1....3....3....1|
|.....................||..................|
=
/...1...................\
|...4......1.............|
|..36.....18......1......|
|.576....432.....68.....1|
|........................|

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A001044, A002190, A061691, A192721, A102221 (row sums), A000275 (alternating row sums).

J := unapply(BesselJ(0, 2*sqrt(-1)*sqrt(z)), z):
G := 1/(1-x*(J(z)-1)):
Gser := simplify(series(G, z = 0, 15)):
for n from 1 to 14 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 14 do seq(coeff(P[n], x, k), k = 1..n) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(
      `if`(n=0, 1, add(x*b(n-i)/i!^2, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)*n!^2):
seq(T(n), n=1..14);  # Alois P. Heinz, Sep 10 2019

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[x b[n-i]/i!^2, {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n] n!^2];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Generating function: Let J(z) = Sum_{n>=0} z^n/n!^2. Then
1 + Sum_{n>=1} (Sum_{k = 1..n} T(n,k)*x^k)*z^n/n!^2 = 1/(1 - x*(J(z) - 1))
  = 1 + x*z + (x + 4*x^2)*z^2/2!^2 + (x + 18*x^2 + 36*x^3)*z^3/3!^2 + ....
Relations with other sequences:
The change of variable z -> z/x followed by x -> 1/(x - 1) transforms the above bivariate generating function 1/(1 - x*(J(z) - 1)) into (1 - x)/(-x + J(z*(x-1))), which is the generating function for A192721.
1/k!*T(n,k) = A061691(n,k).
T(n,n) = n!^2 = A001044(n).
Row sums = A102221.
For n>=1, Sum_{k = 1..n} (-1)^(n+k)*T(n,k)/k = A002190(n).

A336439 a(n) = (n!)^n * [x^n] -log(Sum_{k>=0} (-x)^k / (k!)^n).

Original entry on oeis.org

0, 1, 1, 46, 63111, 4226436876, 21095962423437280, 11165885881625823212655540, 846105231095934499366980692096995455, 11911559696594230804398683820096471009503594129080, 39208751872375132639833577214095359308827747721266594509276656136

Offset: 0

Ilya Gutkovskiy, Jul 21 2020

nonn

Cf. A002190, A193420, A275044, A336437, A336438, A336440.

Table[(n!)^n SeriesCoefficient[-Log[Sum[(-x)^k/(k!)^n, {k, 0, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := If[n == 0, 0, (-1)^(n + 1) - (1/n) Sum[(-1)^(n - j) Binomial[n, j]^k j b[j, k], {j, 1, n - 1}]];a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]

A193420 -log( Sum_{n>=0} (-x)^n/n!^3 ) = Sum_{n>=1} a(n)*x^n/n!^3.

Original entry on oeis.org

1, 3, 46, 1899, 163476, 25333590, 6412369860, 2473269931755, 1379817056827720, 1069150908119474628, 1113779885682143602440, 1518901247410616194635510, 2651993653876241574715172280, 5817640695573490720735010689620

Offset: 1

Paul D. Hanna, Jul 26 2011

nonn

			L(x) = -log(1 - x + x^2/2!^3 - x^3/3!^3 + x^4/4!^3 - x^5/5!^3 +-...)
where
L(x) = x + 3*x^2/2!^3 + 46*x^3/3!^3 + 1899*x^4/4!^3 + 163476*x^5/5!^3 +...
ALTERNATE GENERATING METHOD.
A signed version of A181543(n,k) = C(n,k)^3 begins:
1;
1, 1;
1, 8, 1;
1, 27, 27, 1;
1, 64, 216, 64, 1;
1, 125, 1000, 1000, 125, 1; ...
The matrix log of triangle A181543 begins:
0;
1, 0;
-3, 8, 0;
46, -81, 27, 0;
-1899, 2944, -648, 64, 0;
163476, -237375, 46000, -3000, 125, 0; ...
in which this sequence (signed) is found in column 0.

Cf. A002190 (variant), A181543.

{a(n)=n!^3*polcoeff(-log(sum(m=0,n,(-x)^m/m!^3)+x*O(x^n)),n)}

/* As Column 0 of the Matrix Log of signed Triangle A181543 */
{a(n)=local(L,M=matrix(n+1,n+1,r,c,if(r>=c,(-1)^(r-c)*binomial(r-1,c-1)^3)));
L=sum(n=1,#M,(M^0-M)^n/n);if(n<0,0,L[n+1,1])}

Equals column 0 of the matrix log of triangle T(n,k) = (-1)^(n-k)*C(n,k)^3.

a(n) = -(-1)^n + (1/n) * Sum_{k=1..n-1} (-1)^(n-k-1) * binomial(n,k)^3 * k * a(k). - Ilya Gutkovskiy, Jul 15 2021

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.

cp's OEIS Frontend

A115368 Decimal expansion of first zero of the Bessel function J_0(z).

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A101981 Column 0 of triangle A101980, which is the matrix logarithm of A008459 (squared entries of Pascal's triangle).

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A238390 E.g.f.: x / BesselJ(1, 2*x) (even powers only).

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A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

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A181167 G.f.: 1 = Sum_{n>=0} a(n)*x^n* Sum_{k>=0} C(2n+k,k)^2*(-x)^k.

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A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

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A336439 a(n) = (n!)^n * [x^n] -log(Sum_{k>=0} (-x)^k / (k!)^n).

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A181167 G.f.: 1 = Sum_{n>=0} a(n)x^n Sum_{k>=0} C(2n+k,k)^2*(-x)^k.