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 1-5 of 5 results.

A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

The row polynomials with exponents in increasing order (e.g., third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p. 18, Eq. (7).
Also called Bessel numbers of first kind.
The triangle a(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle a(n-k,k) is A100861, which gives coefficients of scaled Hermite polynomials. - Paul Barry, May 21 2005
Related to k-matchings of the complete graph K_n by a(n,k)=A100861(n+k,k). Related to the Morgan-Voyce polynomials by a(n,k)=(2k-1)!!*A085478(n,k). - Paul Barry, Aug 17 2005
Related to Hermite polynomials by a(n,k)=(-1)^k*A060821(n+k, n-k)/2^n. - Paul Barry, Aug 28 2005
The row polynomials, the Bessel polynomials y(n,x):=Sum_{m=0..n} (a(n,m)*x^m) (called y_{n}(x) in the Grosswald reference) satisfy (x^2)*(d^2/dx^2)y(n,x) + 2*(x+1)*(d/dx)y(n,x) - n*(n+1)*y(n,x) = 0.
a(n-1, m-1), n >= m >= 1, enumerates unordered n-vertex forests composed of m plane (aka ordered) increasing (rooted) trees. Proof from the e.g.f. of the first column Y(z):=1-sqrt(1-2*z) (offset 1) and the Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w). See their remark on p. 28 on plane recursive trees. For m=1 see the D. Callan comment on A001147 from Oct 26 2006. - Wolfdieter Lang, Sep 14 2007
The asymptotic expansions of the higher order exponential integrals E(x,m,n), see A163931 for information, lead to the Bessel numbers of the first kind in an intriguing way. For the first four values of m these asymptotic expansions lead to the triangles A130534 (m=1), A028421 (m=2), A163932 (m=3) and A163934 (m=4). The o.g.f.s. of the right hand columns of these triangles in their turn lead to the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4). The row sums of these four triangles lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four right hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next four right hand columns of A001498. So one by one all columns of the triangle of coefficients of Bessel polynomials appear. - Johannes W. Meijer, Oct 07 2009
a(n,k) also appear as coefficients of (n+1)st degree of the differential operator D:=1/t d/dt, namely D^{n+1}= Sum_{k=0..n} a(n,k) (-1)^{n-k} t^{1-(n+k)} (d^{n+1-k}/dt^{n+1-k}. - Leonid Bedratyuk, Aug 06 2010
a(n-1,k) are the coefficients when expanding (xI)^n in terms of powers of I. Let I(f)(x) := Integral_{a..x} f(t) dt, and (xI)^n := x Integral_{a..x} [ x_{n-1} Integral_{a..x_{n-1}} [ x_{n-2} Integral_{a..x_{n-2}} ... [ x_1 Integral_{a..x_1} f(t) dt ] dx_1 ] .. dx_{n-2} ] dx_{n-1}. Then: (xI)^n = Sum_{k=0..n-1} (-1)^k * a(n-1,k) * x^(n-k) * I^(n+k)(f)(x) where I^(n) denotes iterated integration. - Abdelhay Benmoussa, Apr 11 2025

Examples

			The triangle a(n, k), n >= 0, k = 0..n, begins:
  1
  1  1
  1  3   3
  1  6  15    15
  1 10  45   105    105
  1 15 105   420    945    945
  1 21 210  1260   4725  10395   10395
  1 28 378  3150  17325  62370  135135   135135
  1 36 630  6930  51975 270270  945945  2027025  2027025
  1 45 990 13860 135135 945945 4729725 16216200 34459425 34459425
  ...
And the first few Bessel polynomials are:
  y_0(x) = 1,
  y_1(x) = x + 1,
  y_2(x) = 3*x^2 + 3*x + 1,
  y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1,
  y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1,
  y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1,
  ...
Tree counting: a(2,1)=3 for the unordered forest of m=2 plane increasing trees with n=3 vertices, namely one tree with one vertex (root) and another tree with two vertices (a root and a leaf), labeled increasingly as (1, 23), (2,13) and (3,12). - _Wolfdieter Lang_, Sep 14 2007

References

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

Links

Crossrefs

Cf. A001497 (same triangle but rows read in reverse order). Other versions of this same triangle are given in A144331, A144299, A111924 and A100861.
Columns from left edge include A000217, A050534.
Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, A038121.
Bessel polynomials evaluated at certain x are A001515 (x=1, row sums), A000806 (x=-1), A001517 (x=2), A002119 (x=-2), A001518 (x=3), A065923 (x=-3), A065919 (x=4). Cf. A043301, A003215.
Cf. A245066 (central terms). A113025 (y_n(2*x)).

Programs

  • Haskell
    a001498 n k = a001498_tabl !! n !! k
a001498_row n = a001498_tabl !! n
a001498_tabl = map reverse a001497_tabl
-- Reinhard Zumkeller, Jul 11 2014
  • Magma
    /* As triangle: */ [[Factorial(n+k)/(2^k*Factorial(n-k)*Factorial(k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016
  • Maple
    Bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials
Bessel := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*Bessel(n-1)+Bessel(n-2); fi; end; # recurrence for Bessel polynomials
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;
f := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*f(n-1)+f(n-2); fi; end;
# Alternative:
T := (n,k) -> pochhammer(n+1,k)*binomial(n,k)/2^k:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 11 2018
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (n - k + 1)* T(n, k - 1) + T(n - 1, k) fi fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Oct 02 2023
  • Mathematica
    max=50; Flatten[Table[(n+k)!/(2^k*(n-k)!*k!), {n, 0, Sqrt[2 max]//Ceiling}, {k, 0, n}]][[1 ;; max]] (* Jean-François Alcover, Mar 20 2011 *)
  • PARI
    {T(n,k)=if(k<0||k>n, 0, binomial(n, k)*(n+k)!/2^k/n!)} /* Michael Somos, Oct 03 2006 */
  • PARI
    A001497_ser(N,t='t) = {
  my(x='x+O('x^(N+2)));
  serlaplace(deriv(exp((1-sqrt(1-2*t*x))/t),'x));
};
concat(apply(Vecrev, Vec(A001497_ser(9)))) \\ Gheorghe Coserea, Dec 27 2017

Formula

a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald and Riordan). - Ralf Stephan, Apr 20 2004
a(n, 0)=1; a(0, k)=0, k > 0; a(n, k) = a(n-1, k) + (n-k+1) * a(n, k-1) = a(n-1, k) + (n+k-1) * a(n-1, k-1). - Len Smiley
a(n, m) = A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >= 0, otherwise 0.
G.f. for m-th column: (A001147(m)*x^m)/(1-x)^(2*m+1), m >= 0, where A001147(m) = double factorials (from explicit a(n, m) form).
Row polynomials y_n(x) are given by D^(n+1)(exp(t)) evaluated at t = 0, where D is the operator 1/(1-t*x)*d/dt. - Peter Bala, Nov 25 2011
G.f.: conjecture: T(0)/(1-x), where T(k) = 1 - x*y*(k+1)/(x*y*(k+1) - (1-x)^2/T(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013
Recurrence from Grosswald, p. 18, eq. (5), for the row polynomials: y_n(x) = (2*n-1)*x*y_{n-1} + y_{n-2}(x), y_{-1}(x) = 1 = y_{0} = 1, n >= 1. This becomes, for n >= 0, k = 0..n: a(n, k) = 0 for n < k (zeros not shown in the triangle), a(n, -1) = 0, a(0, 0) = 1 = a(1, 0) and otherwise a(n, k) = (2*n-1)*a(n-1, k-1) + a(n-2, k). Compare with the above given recurrences. - Wolfdieter Lang, May 11 2018
T(n, k) = Pochhammer(n+1,k)*binomial(n,k)/2^k = A113025(n,k)/2^k. - Peter Luschny, May 11 2018
a(n, k) = Sum_{i=0..min(n-1, k)} (n-i)(k-i) * a(n-1, i) where x(n) = x*(x-1)*...*(x-n+1) is the falling factorial, this equality follows directly from the operational formula we wrote in Apr 11 2025.- Abdelhay Benmoussa, May 18 2025

A119274 Triangle of coefficients of numerators in Padé approximation to exp(x).

Original entry on oeis.org

1, 2, 1, 12, 6, 1, 120, 60, 12, 1, 1680, 840, 180, 20, 1, 30240, 15120, 3360, 420, 30, 1, 665280, 332640, 75600, 10080, 840, 42, 1, 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1, 518918400, 259459200, 60540480, 8648640, 831600, 55440, 2520
Offset: 0

Views

Author

Paul Barry, May 12 2006

Keywords

Comments

n-th numerator of Padé approximation is (1/n!)*sum{j=0..n, C(n,j)(2n-j)!x^j}. Reversal of A113025. Row sums are A001517. First column is A001813. Inverse is A119275.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
Dividing each diagonal by its initial element generates A054142. - Tom Copeland, Oct 10 2016

Examples

			Triangle begins
1,
2, 1,
12, 6, 1,
120, 60, 12, 1,
1680, 840, 180, 20, 1,
30240, 15120, 3360, 420, 30, 1

Crossrefs

Cf. A001497, A054142.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (2*n)!/n!, 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    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[(2#)!/#!&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
  • Sage
    # uses[bell_transform from A264428]
# Adds a column 1,0,0,0,... at the left side of the triangle.
def A119274_row(n):
    multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
    mfact = [multifact_4_2(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A119274_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

Formula

Number triangle T(n,k) = C(n,k)(2n-k)!/n!.
After adding a leading column (1,0,0,0,...), the triangle gives the coefficients of the Sheffer associated sequence (binomial-type polynomials) for the delta (lowering) operator D(1-D) with e.g.f. exp[ x * (1 - sqrt(1-4t)) / 2 ] . See Mathworld on Sheffer sequences. See A134685 for relation to Catalan numbers. - Tom Copeland, Feb 09 2008

A168422 Number triangle with row sums given by quadruple factorial numbers A001813.

Original entry on oeis.org

1, 1, 1, 7, 4, 1, 71, 39, 9, 1, 1001, 536, 126, 16, 1, 18089, 9545, 2270, 310, 25, 1, 398959, 208524, 49995, 7120, 645, 36, 1, 10391023, 5394991, 1301139, 190435, 18445, 1197, 49, 1, 312129649, 161260336, 39066076, 5828704, 589750, 41776, 2044, 64, 1
Offset: 0

Views

Author

Paul Barry, Nov 25 2009

Keywords

Comments

Reversal of coefficient array for the polynomials P(n,x) = Sum_{k=0..n} (C(n+k,2k)*(2k)!/k!)*x^k*(1-x)^(n-k).
Note that P(n,x) = Sum_{k=0..n} A113025(n,k)*x^k*(1-x)^(n-k). Row sums are A001813.

Examples

			Triangle begins
          1
          1         1
          7         4        1
         71        39        9       1
       1001       536      126      16      1
      18089      9545     2270     310     25     1
     398959    208524    49995    7120    645    36    1
   10391023   5394991  1301139  190435  18445  1197   49  1
  312129649 161260336 39066076 5828704 589750 41776 2044 64 1
Production matrix begins
        1       1
        6       3       1
       40      20       5      1
      336     168      42      7     1
     3456    1728     432     72     9    1
    42240   21120    5280    880   110   11   1
   599040  299520   74880  12480  1560  156  13  1
  9676800 4838400 1209600 201600 25200 2520 210 15 1
Complete this with a top row (1,0,0,0,...) and invert: we get
    1
   -1   1
   -3  -3   1
   -5  -5  -5   1
   -7  -7  -7  -7   1
   -9  -9  -9  -9  -9   1
  -11 -11 -11 -11 -11 -11   1
  -13 -13 -13 -13 -13 -13 -13   1
  -15 -15 -15 -15 -15 -15 -15 -15   1
  -17 -17 -17 -17 -17 -17 -17 -17 -17   1

Links

Crossrefs

Column 1 is |A002119|.
Sum_{k=0..n} T(n,k) * 2^k, is A001517(n).
Cf. A079267.

Programs

  • PARI
    T(n,k)={sum(j=k, n, (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!))/k!} \\ Andrew Howroyd, Mar 24 2023
  • SageMath
    def T(n,k):
    return(sum((-1)^(j-k) * binomial(2*n-j,n) * binomial(n,j)\
     * binomial(j,k) * factorial(n-j)\
     for j in range(k,n+1))) # William P. Orrick, Mar 24 2023

Formula

T(n,k) = (1/k!)*Sum_{j=k..n} (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!).

Extensions

Corrected and extended by William P. Orrick, Mar 24 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

Views

Author

Wolfdieter Lang, May 07 2018

Keywords

Comments

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

Examples

			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
...

References

  • 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.

Links

Crossrefs

Row sums are A002119.
Cf. A098361, A001498, A113025, A113216.

Programs

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

Formula

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

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 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.