A012244 -id:A012244 - cp's OEIS frontend

A054798 Let N(k) and D(k) be the sequences defined in A054765 and A012244; write N(k)* D(k+j ) - N(k+j)D(k) = (-1)^(k+1)(k!)^2*P(k) where P(k) is a polynomial in k of degree j-1; sequence gives coefficients of expansion of P(k) in powers of k for j=1,2,3,...

1, 2, 3, 5, 20, 19, 12, 90, 214, 160, 29, 348, 1497, 2718, 1744, 70, 1225, 8236, 26453, 40336, 23184, 169, 4056, 39114, 193184, 512813, 689512, 364176, 408, 12852, 167884, 1174860, 4737628, 10955304, 13372072, 6598656, 985, 39400, 669078, 6282340, 35554929, 123708580, 257200712, 290478120, 135484416, 2378, 117711, 2519024, 30514946, 229958030, 1114357079, 3459179856, 6602445344, 6991966752, 3108695040

Offset: 1

N. J. A. Sloane, May 26 2000

			For j=1,2,3 the polynomials P(k) are 1, 3 + 2 k, 19 + 20 k + 5 k^2.
1;
2,3,
5,20,19,
12,90,214,160,
29,348,1497,2718,1744,
70,1225,8236,26453,40336,23184,
169,4056,39114,193184,512813,689512,364176,
408,12852,167884,1174860,4737628,10955304,13372072,6598656,
985,39400,669078,6282340,35554929,123708580,257200712,290478120,135484416,
2378,117711,2519024,30514946,229958030,1114357079,3459179856,6602445344,6991966752,3108695040,

K. S. Brown, Integer Sequences Related To Pi

Sign corrected by R. J. Mathar, Jul 13 2013

A054765 a(n+2) = (2n+3)a(n+1) + (n+1)^2a(n), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 3, 19, 160, 1744, 23184, 364176, 6598656, 135484416, 3108695040, 78831037440, 2189265960960, 66083318415360, 2154235544616960, 75425161203302400, 2822882994841190400, 112463980097804697600

Offset: 0

N. J. A. Sloane, May 26 2000

The denominators of the convergents of [1/3, 4/5, 9/7, 16/9, ...]. To produce Pi the above continued fraction is used. It is formed by n^2/(2*n+1) which starts at n=1. Most numerators of continued fractions are 1 & thus are left out of the brackets. In the case of Pi they vary. Therefore here both numerators & denominators are given. The first 4 convergents are 1/3,5/19,44/160,476/1744. The value of this continued fraction is .273239... . 4*INV(1+.273239...) is Pi. - Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

Starting with offset 1 = row sums of triangle A155729. [Gary W. Adamson & Alexander R. Povolotsky, Jan 25 2009]

G. C. Greubel, Table of n, a(n) for n = 0..390
K. S. Brown, Integer Sequences Related To Pi

Cf. A155729, A012244, A054766.

A054765 := proc(n)
    option remember;
    if n <= 1 then
        n;
    else
        (2*n-1)*procname(n-1)+(n-1)^2*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Jul 13 2013

RecurrenceTable[{a[n + 2] == (2*n + 3)*a[n + 1] + (n + 1)^2*a[n],
a[0] == 0, a[1] == 1}, a, {n,0,50}] (* G. C. Greubel, Feb 18 2017 *)

a(n) ~ Pi * (1+sqrt(2))^(n + 1/2) * n^n / (2^(9/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017

More terms from James Sellers, May 27 2000

A306800 Square array whose entry A(n,k) is the number of endofunctions on a set of size n with preimage constraint {0,1,...,k}, for n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 6, 0, 1, 1, 4, 24, 24, 0, 1, 1, 4, 27, 204, 120, 0, 1, 1, 4, 27, 252, 2220, 720, 0, 1, 1, 4, 27, 256, 3020, 29520, 5040, 0, 1, 1, 4, 27, 256, 3120, 44220, 463680, 40320, 0, 1, 1, 4, 27, 256, 3125, 46470, 765030, 8401680, 362880, 0

Offset: 0

Benjamin Otto, Mar 10 2019

A preimage constraint is a set of nonnegative integers such that the size of the inverse image of any element is one of the values in that set.

Thus, A(n,k) is the number of endofunctions on a set of size n such that each preimage has at most k entries. Equivalently, A(n,k) is the number of n-letter words from an n-letter alphabet such that no letter appears more than k times.

			Array begins:
  1    1     1     1     1 ...
  0    1     1     1     1 ...
  0    2     4     4     4 ...
  0    6    24    27    27 ...
  0   24   204   252   256 ...
  0  120  2220  3020  3120 ...
  0  720 29520 44220 46470 ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.6 and 7.8.

Columns 0-9: A000007, A000142, A012244, A239368, A324804, A324805, A324806, A324807, A324808, A324809.
A(n,n) gives A000312.
Similar array for preimage condition {i>=0 | i!=k}: A245413.
Number of functions with preimage condition given by the even nonnegative integers: A209289.
Sum over all k of the number of functions with preimage condition {0,k}: A231812.
Cf. A019575.

b:= proc(n, i, k) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1, k)*binomial(n, j), j=0..min(k, n))))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Apr 05 2019

b[n_, i_, k_] := b[n, i, k] = If[n==0 && i==0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] Binomial[n, j], {j, 0, Min[k, n]}]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 29 2019, after Alois P. Heinz *)

# print first num_entries entries in column k
import math, sympy; x=sympy.symbols('x')
k=5; num_entries = 64
P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [1]; curr_pow = 1
for term in range(1,num_entries):
   curr_pow=(curr_pow*eP).expand()
   r.append(curr_pow.coeff(x**term)*math.factorial(term))
print(r)

A(n,k) = n! * [x^n] e_k(x)^n, where e_k(x) is the truncated exponential 1 + x + x^2/2! + ... + x^k/k!. When k>1, the link above yields explicit constants c_k, r_k so that the columns are asymptotically c_k * n^(-1/2) * r_k^-n. Stirling's approximation gives column k=1, and column k=0 is 0.

A(n,k) = Sum_{j=1..min(k,n)} A019575(n,j) for n>=1. - Alois P. Heinz, Jun 28 2023

Offset changed to 0 by Alois P. Heinz, Jun 28 2023

A054766 a(n+2) = (2n + 3)a(n+1) + (n + 1)^2*a(n), a(0) = 1, a(1) = 0.

Original entry on oeis.org

1, 0, 1, 5, 44, 476, 6336, 99504, 1803024, 37019664, 849418560, 21539756160, 598194037440, 18056575823040, 588622339549440, 20609136708249600, 771323264354361600, 30729606721005830400, 1298448658633614566400

Offset: 0

N. J. A. Sloane, May 26 2000

Numerators of the convergents of the generalized continued fraction expansion 4/Pi - 1 = [0; 1/3, 4/5, 9/7,..., n^2/(2*n + 1),...] = 1/(3 + 4/(5 + 9/(7 + ...))). The first 4 convergents are 1/3, 5/19, 44/160 and 476/1744.

Indranil Ghosh, Table of n, a(n) for n = 0..392
K. S. Brown, Integer Sequences Related To Pi

Cf. A012244, A054765.

RecurrenceTable[{a[n+2] == (2*n+3)*a[n+1] + (n+1)^2*a[n], a[0] == 1, a[1] == 0}, a, {n, 0, 25}] (* Vaclav Kotesovec, Feb 18 2017 *)
t={1,0};Do[AppendTo[t,(2(n-2)+3)*t[[-1]]+(n-1)^2*t[[-2]]],{n,2,18}];t (* Indranil Ghosh, Feb 25 2017 *)

a(n) ~ (1 - Pi/4) * (1 + sqrt(2))^(n + 1/2) * n^n / (2^(1/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017

More terms from James Sellers, May 27 2000
Definition expanded by Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008
Keyword frac added by Michel Marcus, Feb 25 2017

A098461 Expansion of E.g.f.: 1/sqrt(1-2x-3x^2).

Original entry on oeis.org

1, 1, 6, 42, 456, 6120, 101520, 1980720, 44634240, 1139080320, 32488646400, 1023985670400, 35345049062400, 1325988036172800, 53721616851302400, 2337607853957376000, 108727934847307776000, 5383304681800421376000, 282682783375630589952000

Offset: 0

Paul Barry, Sep 08 2004

G. C. Greubel, Table of n, a(n) for n = 0..379

Cf. A012244, A098460, A002426.

Main diagonal of A094796.

Table[(n!/2^n) Sum[Binomial[n, k] Binomial[2 (n - k), n] 3^k, {k, 0, Floor[n/2]}], {n, 0, 17}] (* Michael De Vlieger, Sep 14 2016 *)

a(n) = (n!/2^n)*A098453(n);
a(n) = (n!/2^n)*Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2*(n-k), n)*3^k.
D-finite with recurrence: a(n) +(1-2n)*a(n-1) -3(n-1)^2*a(n-2)=0. - R. J. Mathar, Dec 11 2011
a(n) = n! * A002426(n). - Anton Zakharov, Sep 14 2016

A190392 E.g.f. A(x) satisfies A'(x) = sin(A(x)) + cos(A(x)).

Original entry on oeis.org

1, 1, 0, -4, -12, 4, 240, 1184, -1008, -59504, -401280, 643136, 38584128, 323581504, -848090880, -51666451456, -509739310848, 2004840714496, 123888658698240, 1386061762251776, -7721141999864832, -483475390212586496, -5974101514137292800, 45231727252157947904

Offset: 1

Vladimir Kruchinin, May 09 2011

sign

Let f(x) be a smooth function. The autonomous differential equation A'(x) = f(A(x)), with initial condition A(0) = 0, is separable and the solution is given by A(x) = inverse function of Integral_{t = 0..x} 1/f(t) dt. The inversion of the integral Integral_{t = 0..x} 1/f(t) dt is most conveniently found by applying [Dominici, Theorem 4.1]. The result is A(x) = Sum_{n>=1} D^(n-1)[f](0)*x^n/n!, where the nested derivative D^n[f](x)is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = (d/dx)(f(x)*D^n[f](x)) for n >= 0. See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x). In the present case we take f(x) = sin(x)+cos(x). - Peter Bala, Aug 27 2011

Alois P. Heinz, Table of n, a(n) for n = 1..200
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.

Cf. A001586, A012244, A028296.

A := x ; for i from 1 to 35 do sin(A)+cos(A) ; convert(taylor(%,x=0,25),polynom) ; A := int(%,x) ; print(A) ; end do:
for i from 1 to 25 do printf("%d,", coeftayl(A,x=0,i)*i!) ; end do: # R. J. Mathar, Jun 03 2011

terms = 25; A[] = 0; Do[A[x] = Integrate[Sin[A[x]] + Cos[A[x]], x] + O[x]^terms, terms]; CoefficientList[A[x], x]*Range[0, terms-1]! (* Jean-François Alcover, Feb 21 2013, updated Jan 15 2018 *)

g(n):=(-1)^floor(n/2)*1/n!;
a(n):=T190015_Solve(n,g);

E.g.f.: A(x) = inverse of Integral_{t = 0..x} 1/(sin(t)+cos(t)) dt = series reversion (x - x^2/2! + 3*x^3/3! - 11*x^4/4! + 57*x^5/5! - ...) = x + x^2/2! - 4*x^4/4! - 12*x^5/5! + .... a(n) = D^(n-1)[sin(x)+cos(x)](0), where the nested derivative operator D^n is defined above. Compare with A012244. -Peter Bala, Aug 27 2011
E.g.f.: A(x) = 2*arctan((sqrt(2)-1)*exp(sqrt(2)*x))-Pi/4. Compare with A028296. - Peter Bala, Sep 02 2011
G.f.: 1/G(0) where G(k) = 1 - 2*x*(k+1)/(1 + 1/(1 - 2*x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 10 2013.
G.f.: -(1/x)/Q(0), where Q(k)= 2*k+1 - 1/x + (k+1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 15 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 + (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013
E.g.f. if offset 0: 2^(1/2)/(2^(1/2)*cosh(x*2^(1/2))-sinh(x*2^(1/2))). - Sergei N. Gladkovskii, Nov 10 2013
a(n) ~ -(n-1)! * 2^(3*n/2+1) * sin(n*arctan(Pi/log(3 - 2*sqrt(2)))) / (Pi^2 + log(3 - 2*sqrt(2))^2)^(n/2). - Vaclav Kotesovec, Jan 07 2014

A089975 Array read by ascending antidiagonals: T(n,k) is the number of n-letter words from a k-letter alphabet such that no letter appears more than twice.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 4, 3, 1, 0, 0, 6, 9, 4, 1, 0, 0, 6, 24, 16, 5, 1, 0, 0, 0, 54, 60, 25, 6, 1, 0, 0, 0, 90, 204, 120, 36, 7, 1, 0, 0, 0, 90, 600, 540, 210, 49, 8, 1, 0, 0, 0, 0, 1440, 2220, 1170, 336, 64, 9, 1, 0, 0, 0, 0, 2520, 8100, 6120, 2226, 504, 81, 10, 1

Offset: 0

Paul Boddington, Nov 17 2003

			Array begins:
  1, 1, 1,  1,    1,     1,      1,      1,       1,       1,       1, ...
  0, 1, 2,  3,    4,     5,      6,      7,       8,       9,      10, ...
  0, 1, 4,  9,   16,    25,     36,     49,      64,      81,     100, ...
  0, 0, 6, 24,   60,   120,    210,    336,     504,     720,     990, ...
  0, 0, 6, 54,  204,   540,   1170,   2226,    3864,    6264,    9630, ...
  0, 0, 0, 90,  600,  2220,   6120,  14070,   28560,   52920,   91440, ...
  0, 0, 0, 90, 1440,  8100,  29520,  83790,  201600,  430920,  842400, ...
  0, 0, 0,  0, 2520, 25200, 128520, 463680, 1345680, 3356640, 7484400, ...
  ... - _Robert FERREOL_, Nov 03 2017

Dennis Walsh, Width-Restricted Finite Functions

T(1, k) = A001477(k); T(2, k) = A000290(k); T(3, k) = A007531(k); T(n, n) = A012244(n); T(n, n+1) = A036774(n); T(n, n+2) = A003692(n+1); T(2*n, n) = A000680(n); sum(T(n, k), n=0..2*k) = A003011(k); sum(T(r, n-r), r=0..n) = A089976(n).

See A141765 for an irregular triangle version : T(n,k)=A141765(k,n) for n <= 2k.

T:=(n,k)->add(n!*k!/(n-2*i)!/i!/(k-n+i)!/2^i,i=max(0,n-k)..n/2):
or
T:=proc(n,k) option remember :if n=0 then 1 elif n=1 then k elif k=0 then 0 else T(n, k-1)+n*T(n-1, k-1)+binomial(n,2)*T(n-2, k-1) fi end:
or
T:=(n,k)-> n!*coeff((1 + x + x^2/2)^k, x,n):
seq(seq(T(n-k,k),k=0..n),n=0..20);
# Robert FERREOL, Nov 07 2017

T[n_, k_] := Sum[n!*k!/(2^i*(n - 2 i)!*(k - n + i)!*i!), {i, Max[0, n - k], Floor[n/2]}];
Table[T[n-k , k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 05 2017, after Robert FERREOL *)

from math import factorial as f
def T(n,k):
    return sum(f(n)*f(k)//f(n-2*i)//f(i)//f(k-n+i)//2**i for i in range(max(0,n-k),n//2+1))
[T(n-k,k) for n in range(21) for k in range(n+1)]
# Robert FERREOL, Oct 17 2017

T(n, k) = T(n, k-1) + n*T(n-1, k-1) + binomial(n, 2)*T(n-2, k-1) for n >= 2 and k >= 1.
T(n, k) = Sum_{i=max(0,n-k)..floor(n/2)} n!*k!/(2^i*(n-2*i)!*(k-n+i)!*i!). - Robert FERREOL, Oct 30 2017
T(n,k) = (-1)^n*n!*2^(-n/2)*GegenbauerC(n, -k, 1/sqrt(2)) for k >= n. - Robert Israel, Nov 08 2017
G.f.: Sum({n>=0} T(n,k)x^n)=n!(1 + x + x^2/2)^k. See Walsh link. - Robert FERREOL, Nov 14 2017

A239368 Number of words of length n over the alphabet {0,...,n-1} that avoid the pattern 1111.

Original entry on oeis.org

1, 1, 4, 27, 252, 3020, 44220, 765030, 15269520, 345376080, 8730489600, 243911883600, 7463164262400, 248207881521600, 8915064168410400, 343923449355486000, 14182674669779616000, 622591172035376544000, 28986699477880400256000, 1426677017904959524704000

Offset: 0

Chad Brewbaker, Mar 17 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A012244, A239295.

a:= proc(n) option remember; `if`(n<3, n^n,
     ((105*n^3-252*n^2+175*n-36) *a(n-1) -2*(n-1)^2 *a(n-2)
     +2*(5*n-2)*(n-1)^2*(n-2)^2*a(n-3)) / (4*(2*n-1)*(5*n-7)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 20 2014

Recursion: see Maple program.

a(8)-a(11) from Alois P. Heinz, Mar 17 2014

a(12)-a(19) from Alois P. Heinz, Jul 20 2014

A345125 Numerator of 4/(1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + ... + (n-1)^2/(2*n-1) )))).

Original entry on oeis.org

0, 4, 3, 19, 160, 1744, 644, 2529, 183296, 3763456, 4317632, 54743776, 1013549056, 30594128896, 35618973952, 10392576224, 3111643512832, 123968232030208, 48501417558016, 1083228572868608, 4080033616887808, 188557135970304, 3781715948011520

Offset: 0

Seiichi Manyama, Sep 16 2021

The limit of a(n)/A345259(n) is Pi.

			4/(1 + 1^2/(3 + 2^2/5)) = 19/6. So a(3) = 19.
0, 4, 3, 19/6, 160/51, 1744/555, 644/205, 2529/805, 183296/58345, ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.

Cf. A012244, A054765, A054766, A345259 (denominator).

nmax = 25; Join[{0}, Table[4/(1 + ContinuedFractionK[j^2, (2*j + 1), {j, 1, k}]), {k, 0, nmax}] // Numerator] (* Vaclav Kotesovec, Sep 16 2021 *)

a(n) = my(x=0); forstep(i=n, 2, -1, x = (i-1)^2/((2*i-1)+x);); if (n, numerator(4/(1+x)), numerator(x)); \\ Michel Marcus, Sep 16 2021

a(n)/A345259(n) = 4 * A054765(n)/A012244(n).

A345259 Denominator of 4/(1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + ... + (n-1)^2/(2*n-1) )))).

Original entry on oeis.org

1, 1, 1, 6, 51, 555, 205, 805, 58345, 1197945, 1374345, 17425485, 322622685, 9738413685, 11337871545, 3308059755, 990466892415, 39460313827935, 15438480702645, 344802363740835, 1298715036217599, 60019600489849, 1203757572990973

Offset: 0

Seiichi Manyama, Sep 16 2021

The limit of A345125(n)/a(n) is Pi.

			4/(1 + 1^2/(3 + 2^2/5)) = 19/6. So a(3) = 6.
0, 4, 3, 19/6, 160/51, 1744/555, 644/205, 2529/805, 183296/58345, ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.

Cf. A012244, A054765, A054766, A345125 (numerator).

nmax = 25; Join[{1}, Table[4/(1 + ContinuedFractionK[j^2, (2*j + 1), {j, 1, k}]), {k, 0, nmax}] // Denominator] (* Vaclav Kotesovec, Sep 16 2021 *)

a(n) = my(x=0); forstep(i=n, 2, -1, x = (i-1)^2/((2*i-1)+x);); if (n, denominator(4/(1+x)), denominator(x)); \\ Michel Marcus, Sep 16 2021

A345125(n)/a(n) = 4 * A054765(n)/A012244(n).

cp's OEIS Frontend

A054798 Let N(k) and D(k) be the sequences defined in A054765 and A012244; write N(k)* D(k+j ) - N(k+j)*D(k) = (-1)^(k+1)*(k!)^2*P(k) where P(k) is a polynomial in k of degree j-1; sequence gives coefficients of expansion of P(k) in powers of k for j=1,2,3,...

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A054765 a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 0, a(1) = 1.

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A306800 Square array whose entry A(n,k) is the number of endofunctions on a set of size n with preimage constraint {0,1,...,k}, for n >= 0, k >= 0, read by descending antidiagonals.

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A054766 a(n+2) = (2*n + 3)*a(n+1) + (n + 1)^2*a(n), a(0) = 1, a(1) = 0.

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A098461 Expansion of E.g.f.: 1/sqrt(1-2*x-3*x^2).

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A190392 E.g.f. A(x) satisfies A'(x) = sin(A(x)) + cos(A(x)).

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A089975 Array read by ascending antidiagonals: T(n,k) is the number of n-letter words from a k-letter alphabet such that no letter appears more than twice.

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A239368 Number of words of length n over the alphabet {0,...,n-1} that avoid the pattern 1111.

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A054798 Let N(k) and D(k) be the sequences defined in A054765 and A012244; write N(k)* D(k+j ) - N(k+j)D(k) = (-1)^(k+1)(k!)^2*P(k) where P(k) is a polynomial in k of degree j-1; sequence gives coefficients of expansion of P(k) in powers of k for j=1,2,3,...

A054765 a(n+2) = (2n+3)a(n+1) + (n+1)^2a(n), a(0) = 0, a(1) = 1.

A054766 a(n+2) = (2n + 3)a(n+1) + (n + 1)^2*a(n), a(0) = 1, a(1) = 0.

A098461 Expansion of E.g.f.: 1/sqrt(1-2x-3x^2).