A001498 -id:A001498 - cp's OEIS frontend

A065923 Bessel polynomial y_n(-3).

1, -2, 19, -287, 6046, -163529, 5402503, -210861146, 9494154073, -484412718869, 27621019129606, -1740608617884047, 120129615653128849, -9011461782602547722, 730048534006459494331, -63523233920344578554519, 5908390803126052265064598

Offset: 0

N. J. A. Sloane, Dec 08 2001

sign

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Polynomial coefficients are in A001498.

f:= gfun:-rectoproc({a(n)= -(6*n-3)*a(n-1) + a(n-2),a(0)=1,a(1)=-2},a(n),remember):
map(f, [$0..50]); # Robert Israel, Aug 16 2017

Table[Pochhammer[1/2, n]*(-6)^n*Hypergeometric1F1[0 - n, -2*n, -2/3], {n, 0, 50}] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,50, print1(sum(k=0,n, ((n+k)!/(k! * (n-k)!))*(-3/2)^k), ", ")) \\ G. C. Greubel, Aug 14 2017

a(n) = -(6*n-3)*a(n-1) + a(n-2) for n >= 2. - Sergei N. Gladkovskii, May 17 2013
G.f.: 1/Q(0), where Q(k)= 1 - x + 3*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
From G. C. Greubel, Aug 14 2017: (Start)
a(n) = (1/2){n} * (-3)^n *hypergeometric1f1(-n; -2n; -2/3), where (a){n} is the Pochhammer symbol.
E.g.f.: (1+6*x)^(-1/2) * exp((sqrt(1+6*x) - 1)/3). (End)
G.f.: (1/(1-x))*hypergeometric2f0(1,1/2; - ; -6*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

A130563 Fourth column (m=3) of the Laguerre-Sonin a=1/2 coefficient triangle.

Original entry on oeis.org

1, 36, 990, 25740, 675675, 18378360, 523783260, 15713497800, 496939367925, 16564645597500, 581419060472250, 21459648959248500, 831561397170879375, 33774185977401870000, 1435402904039579475000, 63731888939357328690000

Offset: 3

Wolfdieter Lang, Jul 13 2007

G. C. Greubel, Table of n, a(n) for n = 3..400

Columns m=0, 1, 2 from A001147, A000457, A001881.

Eighth right hand column of triangle A001498. - Johannes W. Meijer, Oct 16 2009

[Round(Factorial(n)*2^(n-3)*Gamma(n+3/2)/(6*Gamma(n-2)*Gamma(9/2))): n in [3..20]]; // G. C. Greubel, May 12 2018

Table[n!*(2^(n - 3))*Binomial[n + 1/2, n - 3]/3!, {n, 3, 50}] (* G. C. Greubel, May 12 2018 *)

for(n=3, 20, print1(n!*(2^(n-3))*binomial(n+1/2, n-3)/3!, ", ")) \\ G. C. Greubel, May 12 2018

a(n) = n!*(2^(n-3))*binomial(n+1/2,n-3)/3!, n >= 3.
a(n) = binomial(n,3)*(2*n+1)!!/7!!, with (2*n+1)!! = A001147(n+1).
a(n) = -A130757(n,3), n >= 3.

A133480 Left 3-step factorial (n,-3)!: a(n) = (-1)^n * A008544(n).

Original entry on oeis.org

1, -2, 10, -80, 880, -12320, 209440, -4188800, 96342400, -2504902400, 72642169600, -2324549427200, 81359229952000, -3091650738176000, 126757680265216000, -5577337931669504000, 262134882788466688000, -13106744139423334400000, 694657439389436723200000, -38900816605808456499200000

Offset: 0

Tom Copeland, Dec 23 2007

To motivate the definition, consider c(t) = column vector(1, t, t^2, t^3, t^4, t^5, ...), T = A094638 and the list of integers.
Starting at 1 and sampling every integer to the right, we obtain (1,2,3,4,5,...) from which factorials may be formed. It's true that
T * c(1) = (1, 1*2, 1*2*3, 1*2*3*4, ...), giving n! for n > 0. Call this sequence the right 1-step factorial (n,+1)!.
Starting at 1 and sampling every integer to the left, we obtain (1,0,-1,-2,-3,-4,-5,...). And,
T * c(-1) = (1, 1*0, 1*0*-1, 1*0*-1*-2, ...) = (1,0,0,0,...). Call this the left 1-step factorial (n,-1)!.
Sampling every other integer to the right, we obtain (1,3,5,7,9,...).
T * c(2) = (1, 1*3, 1*3*5, ...) = (1,3,15,105,945,...), giving A001147 for n > 0, the right 2-step factorial, (n,+2)!.
Sampling every other integer to the left, we obtain (1,-1,-3,-5,-7,...).
T * c(-2) = (1, 1*-1, 1*-1*-3, 1*-1*-3*-5, ...) = (1,-1,3,-15,105,-945,...) = signed A001147, the left 2-step factorial, (n,-2)!.
Sampling every 3 steps to the right, we obtain (1,4,7,10,...).
T * c(3) = (1, 1*4, 1*4*7, ...) = (1,4,28,280,...), giving A007559 for n > 0, the right 3-step factorial, (n,+3)!.
Sampling every 3 steps to the left, we obtain (1,-2,-5,-8,-11,...), giving
T * c(-3) = (1, 1*-2, 1*-2*-5, 1*-2*-5*-8, ...) = (1,-2,10,-80,880,...) = signed A008544 = the left 3-step factorial, (n,-3)!.
The list partition transform A133314 of [1,T * c(t)] gives signed [1,T *c(-t)]. For example:
LPT[1,T*c(1)] = LPT[1,(n,+1)! ] = LPT[A000142] = (1,-1,0,0,0,...) = signed [1,(n,-1)! ]
LPT[1,T*c(2)] = LPT[1,(n,+2)! ] = LPT[A001147] = (1,-1,-1,-3,-15,-105,-945,...) = (1,-A001147) = signed [1,(n,-2)! ]
LPT[1,T*c(3)] = LPT[1,(n,+3)! ] = LPT[A007559] = (1,-1,-2,-10,-80,-880,...) = (1,-A008544) = signed [1,(n,-3)! ]
LPT[1,T*c(-3)] = LPT[1,(n,-3)! ] = signed A007559 = signed [1,(n,+3)! ].
And, e.g.f.[1,T * c(m)] = e.g.f.[1,(n,m)! ] = (1-m*x)^(-1/m).
Also with P(n,t) = Sum_{k=0..n-1} T(n,k+1) * t^k = 1*(1+t)*(1+2t)...(1+(n-1)*t) and P(0,t)=1, exp[P(.,t)*x] = (1-tx)^(-1/t).
T(n,k+1) = (1/k!) (D_t)^k (D_x)^n [ (1-tx)^(-1/t) - 1 ] evaluated at t=x=0.
And, (1-tx)^(-1/t) - 1 is the e.g.f. for plane increasing m-ary trees when t = (m-1), discussed by Bergeron et al. in "Varieties of Increasing Trees" and the book Combinatorial Species and Tree-Like Structures, cited in the OEIS.
The above relations reveal the intimate connections, through T or LPT or sampling, between the right and left step factorials, (n,+m)! and (n,-m)!. The pairs have conjugate interpretations as trees, ignoring signs, which Callan and Lang have noted in several of the OEIS entries above. Also note unsigned (n,-2)! is the diagonal of A001498 and (n,+2)!, the first subdiagonal.

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A000142, A001147, A001498, A007559, A008544, A133314.

[Round((-3)^n*Gamma(n+2/3)/Gamma(2/3)): n in [0..20]]; // G. C. Greubel, Mar 31 2019

Table[(-3)^n*Pochhammer[2/3, n], {n,0,20}] (* G. C. Greubel, Mar 31 2019 *)

vector(20, n, n--; round((-3)^n*gamma(n+2/3)/gamma(2/3))) \\ G. C. Greubel, Mar 31 2019

[(-3)^n*rising_factorial(2/3,n) for n in (0..20)] # G. C. Greubel, Mar 31 2019

a(n) = b(0)*b(1)...b(n) where b = (1,-2,-5,-8,-11,...) .
a(n) = 3^(n+1)*Sum_{k=1..n+1} stirling1(n+1,k)/3^k. - Vladimir Kruchinin, Jul 02 2011
G.f.: (1/Q(0)-1)/x where Q(k) = 1 + x*(3*k-1)/( 1 + x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 22 2013
G.f.: G(0)/(2*x) - 1/x, where G(k) = 1 + 1/(1 - x*(3*k-1)/(x*(3*k-1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
From G. C. Greubel, Mar 31 2019: (Start)
G.f.: Hypergeometric2F0(1,2/3; -; -3*x).
E.g.f.: (1+3*x)^(-2/3).
a(n) = (-3)^n*Pochhammer(2/3, n) = (-3)^n*(Gamma(n+2/3)/Gamma(2/3)). (End)
D-finite with recurrence: a(n) +(3*n-1)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

Terms a(11) onward added by G. C. Greubel, Mar 31 2019

A043302 Triangular table of 2^n (n+k)! / ((n-k)! k! * 4^k).

Original entry on oeis.org

1, 2, 1, 4, 6, 3, 8, 24, 30, 15, 16, 80, 180, 210, 105, 32, 240, 840, 1680, 1890, 945, 64, 672, 3360, 10080, 18900, 20790, 10395, 128, 1792, 12096, 50400, 138600, 249480, 270270, 135135, 256, 4608, 40320, 221760, 831600, 2162160, 3783780, 4054050

Offset: 1

Michael Somos, Jul 16 2002

			1; 2, 1; 4, 6, 3; 8, 24, 30, 15; 16, 80, 180, 210, 105; ...

Cf. A000079, A001147, A001498, A043301, A043302. Row sums is A043301.

{T(n, k) = if( k<0 || k>n, 0, 2^n * (n+k)! / ((n-k)! * k! * 4^k))}

T(n, 0) = A001147(n). T(n, n) = A000079(n).

A065931 Triangle of coefficients of Bessel polynomials {y_n(x)}'.

Original entry on oeis.org

1, 3, 6, 6, 30, 45, 10, 90, 315, 420, 15, 210, 1260, 3780, 4725, 21, 420, 3780, 18900, 51975, 62370, 28, 756, 9450, 69300, 311850, 810810, 945945, 36, 1260, 20790, 207900, 1351350, 5675670, 14189175, 16216200, 45, 1980, 41580, 540540, 4729725, 28378350, 113513400, 275675400, 310134825

Offset: 1

N. J. A. Sloane, Dec 08 2001

			For n = 1 .. 4 the polynomials are
  (y_{1}(x))' =  1;
  (y_{2}(x))' =  3 +  6*x;
  (y_{3}(x))' =  6 + 30*x +  45*x^2;
  (y_{4}(x))' = 10 + 90*x + 315*x^2 + 420*x^3.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Cf. A001497, A001498, A065943.

f:=Factorial;; Flat(List([1..12], n-> List([0..n-1], k-> (f(n+k+1)/(f(k)*f(n-k-1)))*(1/2)^(k+1) ))); # G. C. Greubel, Jul 10 2019

f:=Factorial; [(f(n+k+1)/(f(k)*f(n-k-1)))*(1/2)^(k+1): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jul 10 2019

Table[(n+k+1)!/(k!*(n-k-1)!)*(1/2)^(k+1), {n,1,12}, {k,0,n-1}]//Flatten (* G. C. Greubel, Jul 10 2019 *)

for(n=1,12, for(k=0,n-1, print1((n+k+1)!/(k!*(n-k-1)!)*(1/2)^(k+1), ", "))) \\ G. C. Greubel, Jul 10 2019

f=factorial; [[(f(n+k+1)/(f(k)*f(n-k-1)))*(1/2)^(k+1) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Jul 10 2019

From G. C. Greubel, Jul 10 2019: (Start)
(y_{n}(x))' = (1/2)*Sum_{k=0..n-1} ((n+k+1)!/(k!*(n-k-1)!))*(x/2)^k.
T(n, k) = ((n+k+1)!/(k!*(n-k-1)!))*(1/2)^(k+1) for 0 <= k <= n-1, n>=1. (End)

A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 15, 15, 0, 1, 10, 45, 105, 105, 0, 1, 15, 105, 420, 945, 945, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 0

Offset: 0

Eric W. Weisstein, Mar 14 2005

			Bessel polynomials begin with:
      x;
      x +     x^2;
    3*x +   3*x^2 +    x^3;
   15*x +  15*x^2 +  6*x^3 +    x^4;
  105*x + 105*x^2 + 45*x^3 + 10*x^4 + x^5;
  ...
Triangle of coefficients begins as:
  0;
  1,  0;
  1,  1    0;
  1,  3,   3     0;
  1,  6,  15,   15      0;
  1, 10,  45,  105,   105      0;
  1, 15, 105,  420,   945,   945       0;
  1, 21, 210, 1260,  4725, 10395,  10395       0;
  1, 28, 378, 3150, 17325, 62370, 135135, 135135    0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Bessel Polynomial

Essentially the same as A001498 (the main entry).

Cf. A000085, A000806, A001464, A001515.

A104548:= func< n,k | k eq n select 0 else Binomial(n-1,k)*Factorial(n+k-1)/(2^k*Factorial(n-1)) >;
[A104548(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 02 2023

T[n_, k_]:= If[k==n, 0, Binomial[n-1,k]*(n+k-1)!/(2^k*(n-1)!)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 02 2023 *)

def A104548(n,k): return 0 if (k==n) else binomial(n-1,k)*factorial(n+k-1)/(2^k*factorial(n-1))
flatten([[A104548(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 02 2023

From G. C. Greubel, Jan 02 2023: (Start)
T(n, k) = binomial(n-1,k)*(n+k-1)!/(2^k*(n-1)!), with T(n, n) = 0.
Sum_{k=0..n} T(n, k) = A001515(n-1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000806(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000085(n-1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A001464(n-1). (End)

T(0, 0) = 0 prepended by G. C. Greubel, Jan 02 2023

A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.

Original entry on oeis.org

1, 1, -2, 1, -6, 12, 1, -12, 60, -120, 1, -20, 180, -840, 1680, 1, -30, 420, -3360, 15120, -30240, 1, -42, 840, -10080, 75600, -332640, 665280, 1, -56, 1512, -25200, 277200, -1995840, 8648640, -17297280, 1, -72, 2520, -55440, 831600, -8648640, 60540480, -259459200, 518918400, 1, -90, 3960, -110880, 2162160, -30270240, 302702400, -2075673600, 8821612800, -17643225600, 1, -110, 5940, -205920, 5045040, -90810720, 1210809600, -11762150400, 79394515200, -335221286400, 670442572800

Offset: 0

Wolfdieter Lang, May 07 2018

The Niven potentials N(n, x) = (1/n!)*x^n*(1 - x)^n = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!), with (n-k)!*k! = A098361(n, k), are used to prove the irrationality of Pi^2 (hence Pi). See the Niven and Havil references.

The row polynomials R(n, x) = Sum_{k=0..n} T(n, k) *x^k are R(n, x) = y_n(-2*x), with the Bessel polynomials of Krall and Frink y_n(x) with coefficients given in A001498. There the references are given. - Wolfdieter Lang, May 12 2018

			The triangle T(n, k) begins:
n\k 0   1    2      3      4        5        6          7         8 ...
0:  1
1:  1  -2
2:  1  -6   12
3:  1 -12   60   -120
4:  1 -20  180   -840   1680
5:  1 -30  420  -3360  15120   -30240
6:  1 -42  840 -10080  75600  -332640    66528
7:  1 -56 1512 -25200 277200 -1995840  8648640  -17297280
8:  1 -72 2520 -55440 831600 -8648640 60540480 -259459200 518918400
...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.

Muniru A Asiru, Rows n = 0..50

Row sums are A002119.

Cf. A098361, A001498, A113025, A113216.

Flat(List([0..10],n->List([0..n],k->(-1)^k*Binomial(n,n-k)*Factorial(n+k)/Factorial(n)))); # Muniru A Asiru, May 15 2018

T := (n, k) -> (-1)^k*pochhammer(n+1, k)*binomial(n, k):
seq(print(seq(T(n, k), k=0..n)), n=0..9); # Peter Luschny, May 11 2018

T(n,k)=(-1)^k*binomial(n,n-k)*binomial(n+k,n)*k! \\ M. F. Hasler, May 09 2018

T(n, k) = (d/dx)^(n+k) N(n, x) |_{x=0} =: N^{(n+k)}(n, 0), with N(n, x) = (1/n!)*x^n*(1 - x)^n, for n >= 0, k = 0..n.
N^{(n+k)}(n, 1) = (-1)^(n+k)*T(n, k), which has for even n the unsigned rows, and for odd n the unsigned row entries with negative signs.
T(n, k) = (-1)^k*binomial(n, n-k)*((n+k)!/n!).
T(n, k) = (-1)^k*A113025(n,k) with A113025(n,k) = (n+k)!/(k!*(n-k)!) = abs(A113216(n,k)). - M. F. Hasler, May 09 2018
T(n, k) = (-1)^k*Pochhammer(n+1, k)*binomial(n, k). - Peter Luschny, May 11 2018
Recurrence: from the one of the row polynomials R(n, x) = y_n(-2*x): R(n, x) = -2*(2*n-1)*x*R(n-1, x) + R(n-2, x), with R(-1, x) = 1 = R(0, x) = 1, n >= 1 (see A001498), this becomes, for n >= 0, k = 0..n:
  T(n, k) = 0 for n < k, T(n, -1) = 0, T(0, 0) = 1 = T(1, 0) and otherwise
  T(n, k) = -2*(2*n-1)*T(n-1, k-1) + T(n-2, k). - Wolfdieter Lang, May 12 2018

A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

Original entry on oeis.org

1, 1, 0, 3, 0, -1, 15, 0, -6, 0, 105, 0, -45, 0, 1, 945, 0, -420, 0, 15, 0, 10395, 0, -4725, 0, 210, 0, -1, 135135, 0, -62370, 0, 3150, 0, -28, 0, 2027025, 0, -945945, 0, 51975, 0, -630, 0, 1, 34459425, 0, -16216200, 0, 945945, 0, -13860, 0, 45, 0, 654729075, 0, -310134825, 0, 18918900, 0, -315315, 0, 1485, 0, -1

Offset: 0

G. C. Greubel, May 12 2020, following a suggestion from Michel Marcus

Lambert's numerator polynomials related to convergents of tan(x), g(n, x), are given in A334824.

			Polynomials:
f(0, x) = 1;
f(1, x) = x;
f(2, x) = 3*x^2 - 1;
f(3, x) = 15*x^3 - 6*x;
f(4, x) = 105*x^4 - 45*x^2 + 1;
f(5, x) = 945*x^5 - 420*x^3 + 15*x;
f(6, x) = 10395*x^6 - 4725*x^4 + 210*x^2 - 1;
f(7, x) = 135135*x^7 - 62370*x^5 + 3150*x^3 - 28*x;
f(8, x) = 2027025*x^8 - 945945*x^6 + 51975*x^4 - 630*x^2 + 1.
Triangle of coefficients begins as:
        1;
        1, 0;
        3, 0,      -1;
       15, 0,      -6, 0;
      105, 0,     -45, 0,     1;
      945, 0,    -420, 0,    15, 0;
    10395, 0,   -4725, 0,   210, 0,   -1;
   135135, 0,  -62370, 0,  3150, 0,  -28, 0;
  2027025, 0, -945945, 0, 51975, 0, -630, 0, 1.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094674, A334824.

Columns k: A001147 (k=0), A001879 (k=2), A001880 (k=4), A038121 (k=6).

C := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k)*Factorial(n-k)) ) >;
[T(n,k): k in [0..n], n in [0..10]];

Maple

T:= (n, k) -> I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!); seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10);

Mathematica

(* First program *) y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x]; f[n_, k_]:= Coefficient[((-I)^n/2)*(y[n, I*x] + (-1)^n*y[n, -I*x]), x, k]; Table[f[n, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Second program *) Table[ I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten

Sage

[[ i^k*factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*factorial(k)*factorial(n-k)) for k in (0..n)] for n in (0..10)]

Formula

Equals the coefficients of the polynomials, f(n, x), defined by: (Start)

f(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k)!/((2*k)!*(n-2*k)!))*(x/2)^(n-2*k).

f(n, x) = ((2*n)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 1/2, -n, -n+1/2; -1/x^2).

f(n, x) = ((-i)^n/2)*(y(n, i*x) + (-1)^n*y(n, -i*x)), where y(n, x) are the Bessel Polynomials.

f(n, x) = (2*n-1)*x*f(n-1, x) - f(n-2, x).

E.g.f. of f(n, x): cos((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

f(n, 1) = (-1)^n*f(n, -1) = A053983(n) = (-1)^(n+1)*A053984(-n-1) = (-1)^(n+1) * g(-n-1, 1).

f(n, 2) = (-1)^n*f(n, -2) = A053988(n+1). (End)

As a number triangle:

T(n, k) = i^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), where i = sqrt(-1).

T(n, 0) = A001147(n).

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A065923 Bessel polynomial y_n(-3).

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A130563 Fourth column (m=3) of the Laguerre-Sonin a=1/2 coefficient triangle.

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A133480 Left 3-step factorial (n,-3)!: a(n) = (-1)^n * A008544(n).

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A043302 Triangular table of 2^n *(n+k)! / ((n-k)! * k! * 4^k).

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A065931 Triangle of coefficients of Bessel polynomials {y_n(x)}'.

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A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

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A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.

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A043302 Triangular table of 2^n (n+k)! / ((n-k)! k! * 4^k).