cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing order).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 15, 6, 1, 105, 105, 45, 10, 1, 945, 945, 420, 105, 15, 1, 10395, 10395, 4725, 1260, 210, 21, 1, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 34459425, 34459425, 16216200, 4729725, 945945, 135135, 13860, 990, 45, 1
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The (reverse) Bessel polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^m, the row polynomials, called Theta_n(x) in the Grosswald reference, solve x*(d^2/dx^2)P(n,x) - 2*(x+n)*(d/dx)P(n,x) + 2*n*P(n,x) = 0.
With the related Sheffer associated polynomials defined by Carlitz as
B(0,x) = 1
B(1,x) = x
B(2,x) = x + x^2
B(3,x) = 3 x + 3 x^2 + x^3
B(4,x) = 15 x + 15 x^2 + 6 x^3 + x^4
... (see Mathworld reference), then P(n,x) = 2^n * B(n,x/2) are the Sheffer polynomials described in A119274. - Tom Copeland, Feb 10 2008
Exponential Riordan array [1/sqrt(1-2x), 1-sqrt(1-2x)]. - Paul Barry, Jul 27 2010
From Vladimir Kruchinin, Mar 18 2011: (Start)
For B(n,k){...} the Bell polynomial of the second kind we have
B(n,k){f', f'', f''', ...} = T(n-1,k-1)*(1-2*x)^(k/2-n), where f(x) = 1-sqrt(1-2*x).
The expansions of the first few rows are:
1/sqrt(1-2*x);
1/(1-2*x)^(3/2), 1/(1-2*x);
3/(1-2*x)^(5/2), 3/(1-2*x)^2, 1/(1-2*x)^(3/2);
15/(1-2*x)^(7/2), 15/(1-2*x)^3, 6/(1-2*x)^(5/2), 1/(1-2*x)^2. (End)
Also the Bell transform of A001147 (whithout column 0 which is 1,0,0,...). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
Antidiagonals of A099174 are rows of this entry. Dividing each diagonal by its first element generates A054142. - Tom Copeland, Oct 04 2016
The row polynomials p_n(x) of A107102 are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials above, e.g., (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - Tom Copeland, Oct 10 2016
a(n-1,m-1) counts rooted unordered binary forests with n labeled leaves and m roots. - David desJardins, Feb 23 2019
From Jianing Song, Nov 29 2021: (Start)
The polynomials P_n(x) = Sum_{k=0..n} T(n,k)*x^k satisfy: P_n(x) - (d/dx)P_n(x) = x*P_{n-1}(x) for n >= 1.
{P(n,x)} are related to the Fourier transform of 1/(1+x^2)^(n+1) and x/(1+x^2)^(n+2):
(i) For n >= 0, real number t, we have Integral_{x=-oo..oo} exp(-i*t*x)/(1+x^2)^(n+1) dx = Pi/(2^n*n!) * P_n(|t|) * exp(-|t|);
(ii) For n >= 0, real number t, we have Integral_{x=-oo..oo} x*exp(-i*t*x)/(1+x^2)^(n+2) dx = Pi/(2^(n+1)*(n+1)!) * ((-t)*P_n(-|t|)) * exp(-|t|). (End)
Suppose that f(x) is an n-times differentiable function defined on (a,b) for 0 <= a < b <= +oo, then for n >= 1, the n-th derivative of f(sqrt(x)) on (a^2,b^2) is Sum_{k=1..n} ((-1)^(n-k)*T(n-1,k-1)*f^(k)(sqrt(x))) / (2^n*x^(n-(k/2))), where f^(k) is the k-th derivative of f. - Jianing Song, Nov 30 2023

Examples

			Triangle begins
        1,
        1,       1,
        3,       3,      1,
       15,      15,      6,      1,
      105,     105,     45,     10,     1,
      945,     945,    420,    105,    15,    1,
    10395,   10395,   4725,   1260,   210,   21,   1,
   135135,  135135,  62370,  17325,  3150,  378,  28,  1,
  2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1
Production matrix begins
       1,      1,
       2,      2,      1,
       6,      6,      3,     1,
      24,     24,     12,     4,     1,
     120,    120,     60,    20,     5,    1,
     720,    720,    360,   120,    30,    6,   1,
    5040,   5040,   2520,   840,   210,   42,   7,  1,
   40320,  40320,  20160,  6720,  1680,  336,  56,  8, 1,
  362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
This is the exponential Riordan array A094587, or [1/(1-x),x], beheaded.
- _Paul Barry_, Mar 18 2011

References

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

Links

Crossrefs

Reflected version of A001498 which is considered the main entry.
Other versions of this same triangle are given in A144299, A111924 and A100861.
Row sums give A001515. a(n, 0)= A001147(n) (double factorials).
Cf. A104556 (matrix inverse). A039683, A122850.
Cf. A245066 (central terms).
Cf. A054142, A099174, A107102.

Programs

  • Haskell
    a001497 n k = a001497_tabl !! n !! k
a001497_row n = a001497_tabl !! n
a001497_tabl = [1] : f [1] 1 where
   f xs z = ys : f ys (z + 2) where
     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [z, z-1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Jul 11 2014
  • Magma
    /* As triangle */ [[Factorial(2*n-k)/(Factorial(k)*Factorial(n-k)*2^(n-k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 12 2015
  • Maple
    f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
row := n -> seq(coeff(f(n), x, n - k), k = 0..n): seq(row(n), n = 0..9);
  • Mathematica
    m = 9; Flatten[ Table[(n + k)!/(2^k*k!*(n - k)!), {n, 0, m}, {k, n, 0, -1}]] (* Jean-François Alcover, Sep 20 2011 *)
y[n_, x_] := Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n-1/2, 1/x]; t[n_, k_] := Coefficient[y[n, x], x, k]; Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 01 2013 *)
  • PARI
    T(k, n) = if(n>k||k<0||n<0,0,(2*k-n)!/(n!*(k-n)!*2^(k-n))) /* Ralf Stephan */
  • PARI
    {T(n, k) = if( k<0 || k>n, 0, binomial(n, k)*(2*n-k)!/2^(n-k)/n!)}; /* Michael Somos, Oct 03 2006 */
  • Sage
    # uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A001147(n), 9) # Peter Luschny, Jan 19 2016

Formula

a(n, m) = (2*n-m)!/(m!*(n-m)!*2^(n-m)) if n >= m >= 0 else 0 (from Grosswald, p. 7).
a(n, m)= 0, n= m >= 0 (from Grosswald p. 23, (19)).
E.g.f. for m-th column: ((1-sqrt(1-2*x))^m)/(m!*sqrt(1-2*x)).
G.f.: 1/(1-xy-x/(1-xy-2x/(1-xy-3x/(1-xy-4x/(1-.... (continued fraction). - Paul Barry, Jan 29 2009
T(n,k) = if(k<=n, C(2n-k,2(n-k))*(2(n-k)-1)!!,0) = if(k<=n, C(2n-k,2(n-k))*A001147(n-k),0). - Paul Barry, Mar 18 2011
Row polynomials for n>=1 are given by 1/t*D^n(exp(x*t)) evaluated at x = 0, where D is the operator 1/(1-x)*d/dx. - Peter Bala, Nov 25 2011
The matrix product A039683*A008277 gives a signed version of this triangle. Dobinski-type formula for the row polynomials: R(n,x) = (-1)^n*exp(x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-2*(n-1))*(-x)^k/k!. Cf. A122850. - Peter Bala, Jun 23 2014

A119275 Inverse of triangle related to Padé approximation of exp(x).

Original entry on oeis.org

1, -2, 1, 0, -6, 1, 0, 12, -12, 1, 0, 0, 60, -20, 1, 0, 0, -120, 180, -30, 1, 0, 0, 0, -840, 420, -42, 1, 0, 0, 0, 1680, -3360, 840, -56, 1, 0, 0, 0, 0, 15120, -10080, 1512, -72, 1, 0, 0, 0, 0, -30240, 75600, -25200, 2520, -90, 1, 0, 0, 0, 0, 0, -332640, 277200, -55440, 3960, -110, 1
Offset: 0

Views

Author

Paul Barry, May 12 2006

Keywords

Comments

Inverse of A119274.
Row sums are (-1)^(n+1)*A000321(n+1).
Bell polynomials of the second kind B(n,k)(1,-2). - Vladimir Kruchinin, Mar 25 2011
Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) giving unsigned values and adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

Examples

			Triangle begins
1,
-2, 1,
0, -6, 1,
0, 12, -12, 1,
0, 0, 60, -20, 1,
0, 0, -120, 180, -30, 1,
0, 0, 0, -840, 420, -42, 1,
0, 0, 0, 1680, -3360, 840, -56, 1,
0, 0, 0, 0, 15120, -10080, 1512, -72, 1
Row 4: D(x^4) = (1 - x*(d/dx)^2 + x^2/2!*(d/dx)^4 - ...)(x^4) = x^4 - 12*x^3 + 12*x^2.

Links

Crossrefs

Cf. A059344 (unsigned row reverse).
Cf. A034839, A001813, A001815, A086645, A098158, A130561, A231846.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<2,(n+1)*(-1)^n,0), 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    Table[(-1)^(n - k) (n - k)!*Binomial[n + 1, k + 1] Binomial[k + 1, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 12 2016 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[#<2, (#+1) (-1)^#, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
  • Sage
    # uses[inverse_bell_matrix from A265605]
# Unsigned values and an additional first column (1,0,0, ...).
multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_4_2, 9) # Peter Luschny, Dec 31 2015

Formula

T(n,k) = [k<=n]*(-1)^(n-k)*(n-k)!*C(n+1,k+1)*C(k+1,n-k).
From Peter Bala, May 07 2012: (Start)
E.g.f.: exp(x*(t-t^2)) - 1 = x*t + (-2*x+x^2)*t^2/2! + (-6*x^2+x^3)*t^3/3! + (12*x^2-12*x^3+x^4)*t^4/4! + .... Cf. A059344. Let D denote the operator sum {k >= 0} (-1)^k/k!*x^k*(d/dx)^(2*k). The n-th row polynomial R(n,x) = D(x^n) and satisfies the recurrence equation R(n+1,x) = x*R(n,x)-2*n*x*R(n-1,x). The e.g.f. equals D(exp(x*t)).
(End)
From Tom Copeland, Oct 11 2016: (Start)
With initial index n = 1 and unsigned, these are the partition row polynomials of A130561 and A231846 with c_1 = c_2 = x and c_n = 0 otherwise. The first nonzero, unsigned element of each diagonal is given by A001813 (for each row, A001815) and dividing along the corresponding diagonal by this element generates A098158 with its first column removed (cf. A034839 and A086645).
The n-th polynomial is generated by (x - 2y d/dx)^n acting on 1 and then evaluated at y = x, e.g., (x - 2y d/dx)^2 1 = (x - 2y d/dx) x = x^2 - 2y evaluated at y = x gives p_2(x) = -2x + x^2.
(End)

A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 45, 23, 6, 1, 0, 585, 275, 65, 10, 1, 0, 9945, 4435, 990, 145, 15, 1, 0, 208845, 89775, 19285, 2730, 280, 21, 1, 0, 5221125, 2183895, 456190, 62965, 6370, 490, 28, 1, 0, 151412625, 62002395, 12676265, 1715490, 171255, 13230, 798, 36, 1
Offset: 0

Views

Author

Peter Luschny, Dec 30 2015

Keywords

Examples

			[1],
[0, 1],
[0, 1, 1],
[0, 5, 3, 1],
[0, 45, 23, 6, 1],
[0, 585, 275, 65, 10, 1],
[0, 9945, 4435, 990, 145, 15, 1],
[0, 208845, 89775, 19285, 2730, 280, 21, 1],

Links

Crossrefs

Cf. A007696, A264428, A264429.
Bell transforms of other multifactorials are: A000369, A004747, A039683, A051141, A051142, A119274, A132062, A132393, A203412.

Programs

  • Mathematica
    (* The function BellMatrix is defined in A264428. *)
rows = 10;
M = BellMatrix[Pochhammer[1/4, #] 4^# &, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 23 2019 *)
  • Sage
    # uses[bell_transform from A264428]
def A265606_row(n):
    multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
    mfact = [multifact_4_1(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A265606_row(n) for n in (0..7)]

A113216 Triangle of polynomials P(n,x) of degree n related to Pi (see comment) and derived from Padé approximation to exp(x).

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
Offset: 0

Views

Author

Benoit Cloitre, Jan 07 2006

Keywords

Comments

P(n,x) is a sequence of polynomials of degree n with integer coefficients, having exactly n real roots, such that r(n) the smallest root (in absolute value) converges quickly to Pi/2. e.g. the appropriate root for P(5,x) is r(5)=1.5707963(4026....) . To see the rapidity of convergence it is relevant noting that (r(n)-Pi/2)(2n)! -->0 as n grows.

Examples

			P(5,x) = x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240.
Triangle begins:
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;
...

Crossrefs

Cf. A113025 (unsigned variant), A048854, A059344, A119274.

Programs

  • PARI
    P(n,x)=if(n<2,if(n%2,x+2,1),(4*n-2)*P(n-1,x)-x^2*P(n-2,x))
  • PARI
    P(n,x)=sum(i=0,n,x^i*(-1)^floor(i/2)/(n-i)!/i!*(2*n-i)!)

Formula

P(0, x) = 1, P(1, x) = x+2, P(n, x) = (4*n-2)*P(n-1, x)-x^2*P(n-2, x).
P(n, x) = Sum_{0<=i<=n} (-1)^floor(i/2)*(2n-i)!/i!/(n-i)!*x^i.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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