A001193 -id:A001193 - cp's OEIS frontend

A167584 The ED4 array read by antidiagonals.

1, 2, 1, 13, 6, 1, 76, 41, 10, 1, 789, 372, 93, 14, 1, 7734, 4077, 1020, 169, 18, 1, 110937, 53106, 13269, 2212, 269, 22, 1, 1528920, 795645, 198990, 33165, 4140, 393, 26, 1, 28018665, 13536360, 3383145, 563850, 70485, 6996, 541, 30, 1

Offset: 1

Johannes W. Meijer, Nov 10 2009

The coefficients in the upper right triangle of the ED4 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED4 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).
For the ED1, ED2 and ED3 arrays see A167546, A167560 and A167572.
The Madhava-Gregory-Leibniz series representation for Pi/4 is the case m = 0 of the following more general result: for m = 0,1,2,... there holds 1/(2*m)! * Pi/4 = Sum_{k >= 0} ( (-1)^(m+k) * 1/Product_{j = -m .. m} (2*k + 1 + 2*j) ). The entries of this table are given by truncating these series to n-1 terms and then scaling by certain double factorials -- see the formula below. - Peter Bala, Nov 06 2016

			The ED4 array begins with:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1
  2, 6, 10, 14, 18, 22, 26, 30, 34, 38
  13, 41, 93, 169, 269, 393, 541, 713, 909, 1129
  76, 372, 1020, 2212, 4140, 6996, 10972, 16260, 23052, 31540
  789, 4077, 13269, 33165, 70485, 133869, 233877, 382989, 595605, 888045
  7734, 53106, 198990, 563850, 1339110, 2812194, 5389566, 9619770, 16216470, 26081490
  ...
From _Peter Bala_, Nov 06 2016: (Start)
Table extended to nonpositive values of m:
  n\m|     -4     -3    -2    -1    0
  -----------------------------------
   0 |      0      0     0     0    0
   1 |      1      1     1     1    1
   2 |    -18    -14   -10    -6   -2
   3 |    233    141    73    29    9
   4 |  -2844  -1428  -620  -228  -60
   5 |  39309  17877  7149  2325  525
  ...
Column  0: (-1)^(n+1)*(2*n - 3)!!*n. See A001193;
Column -1: (-1)^n*(2*n - 5)!!/3!!*n*(7 - 4*n^2);
Column -2: (-1)^n*(2*n - 7)!!/5!!*n(-149 + 120*n^2 - 16*n^4);
Column -3: (-1)^n*(2*n - 9)!!/7!!*n*(6483 - 6076*n^2 + 1232*n^4 - 64*n^6);
Column -4: (-1)^n*(2*n - 11)!!/9!!*n*(-477801 + 489136*n^2 - 120288*n^4 + 9984*n^6 - 256*n^8). (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.
Wikipedia, Double factorial

A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the array.
A024199, A167588 and A167589 equal the first three columns of the array.
A167590 equals the row sums of the ED4 array read by antidiagonals.
A167591 is a triangle related to the a(n) formulas of the rows of the ED4 array.
A167594 is a triangle related to the GF(z) formulas of the rows of the ED4 array.
Cf. A002866 (the 2^(n-1)*n! factor).
Cf. A167546 (ED1 array), A167560 (ED2 array), A167572 (ED3 array). Cf. A001193, A003881.

T := proc (n, m) option remember;
      if n = 0 then 0
       elif n = 1 then 1
       else (4*m-2)*T(n-1,m)+(2*n+2*m-5)*(2*n-2*m-1)*T(n-2,m)
      end if;
     end proc:
#square array read by antidiagonals
seq(seq(T(n-m,m), m = 1..n-1), n = 1..10);
# Peter Bala, Nov 06 2016

T[0, k_] := 0; T[1, k_] := 1; T[n_, k_] := T[n, k] = (4*k - 2)*T[n - 1, k] + (2*n + 2*k - 5)*(2*n - 2*k - 1)*T[n - 2, k]; Table[T[n - k, k], {n, 2, 12}, {k, 1, n - 1}] (* G. C. Greubel, Jan 20 2017 *)

a(n,m) = ((2*m-3)!!/(2*(2*m-2*n-3)!!))*Integral_{y=0..oo} sinh(y*(2*n))/(cosh(y))^(2*m-1) dy for m>n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
Sum_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k) = 2^(n-1)*n!
From Peter Bala, Nov 06 2016: (Start)
T(n,m) = ((2*m - 3)!!/(2*(2*m - 2*n - 3)!!)) * Sum_{k = 0..n-1} (-1)^(k+1)*binomial(2*n - k - 1, k)*2^(2*n - 2*k - 1)*1/(2*n - 2*m - 2*k + 1), for n and m >= 0.
Note the double factorial for a negative odd integer N is defined in terms of the gamma function as N!! = 2^((N+1)/2)*Gamma(N/2 + 1)/sqrt(Pi).
T(n, m) = (2*m - 3)!! * (2*n + 2*m - 3)!! * Sum_{k = 0..n-1} ( (-1)^(m + k + 1) / Product_{j = -(m-1) .. m-1} (2*k + 1 + 2*j) ).
Using this result we can extend the table to nonpositive values of m (the column index). Column 0 is a signed version of A001193. We have for m <= 0, T(n,m) = (2*n - 2*|m| - 3)!!/(2*|m| + 1)!! * Sum_{k = 0..n-1} (-1)^k*Product_{j = -|m|..|m|} (2*k + 1 + 2*j).
Recurrence: T(n, m) = (4*m - 2)*T(n-1, m) + (2*n + 2*m - 5)*(2*n - 2*m - 1)*T(n-2, m).
For a fixed value of n, the entries in row n are polynomial in the value of the column index m. The first few polynomials are [1, 4*m - 2, 12*m^2 - 8*m + 9, 32*m^3 - 16*m^2 + 120*m - 60, 80*m^4 + 952*m^2 - 768*m + 525, ...]. (End)

A167591 A triangle related to the a(n) formulas of the rows of the ED4 array A167584.

Original entry on oeis.org

1, 4, -2, 12, -8, 9, 32, -16, 120, -60, 80, 0, 952, -768, 525, 192, 160, 5664, -5008, 12396, -5670, 448, 896, 27888, -20672, 162740, -133128, 72765, 1024, 3584, 120064, -46720, 1537216, -1562464, 2557296, -1081080, 2304, 12288, 467712, 76800

Offset: 1

Johannes W. Meijer, Nov 10 2009

The a(n) formulas given below correspond to the first ten rows of the ED4 array A167584.

The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

			Row 1: a(n) = 1.
Row 2: a(n) = 4*n - 2.
Row 3: a(n) = 12*n^2 - 8*n + 9.
Row 4: a(n) = 32*n^3 - 16*n^2 + 120*n - 60.
Row 5: a(n) = 80*n^4 + 0*n^3 + 952*n^2 - 768*n + 525.
Row 6: a(n) = 192*n^5 + 160*n^4 + 5664*n^3 - 5008*n^2 + 12396*n - 5670.
Row 7: a(n) = 448*n^6 + 896*n^5 + 27888*n^4 - 20672*n^3 + 162740*n^2 - 133128*n + 72765.
Row 8: a(n) = 1024*n^7 + 3584*n^6 + 120064*n^5 - 46720*n^4 + 1537216*n^3 - 1562464*n^2 + 2557296*n - 1081080.
Row 9: a(n) = 2304*n^8 + 12288*n^7 + 467712*n^6 + 76800*n^5 + 11589216*n^4 - 12058368*n^3 + 47963568*n^2 - 38278080*n + 18243225.
Row 10: a(n) = 5120*n^9 + 38400*n^8 + 1686528*n^7 + 1540608*n^6 + 73898880*n^5 - 66179520*n^4 + 631348672*n^3 - 669559008*n^2 + 869709780*n - 344594250.

A167584 is the ED4 array.
A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the ED4 array.
A001787, A167592, A167593, A168307 and A168308 equal the first five left hand triangle columns.
A001193 equals the first right hand triangle column.
A024199 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Comment and formulas added by Johannes W. Meijer, Nov 23 2009

A059366 Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 15, 9, 9, 15, 105, 60, 54, 60, 105, 945, 525, 450, 450, 525, 945, 10395, 5670, 4725, 4500, 4725, 5670, 10395, 135135, 72765, 59535, 55125, 55125, 59535, 72765, 135135, 2027025, 1081080, 873180, 793800, 771750, 793800, 873180

Offset: 0

N. J. A. Sloane, Jan 28 2001

From Petros Hadjicostas, May 12 2020: (Start)
Following Comtet (1974, pp. 166-167), let J(m) = Integral_{t = 0..Pi/2} (A^2*cos^2(t) + B^2*sin^2(t))^(-m)) dt for m >= 0. Then J(m+1) = (Pi/(2^(m+1)*A*B*m!)) * Sum_{s=0..m} T(m,s)*A^(-2*s)*B^(-2*m+2*s).
Given m >= 0, the collection of numbers T(m,s)/A000165(m) = T(m,s)/(m!*2^m), s = 0..m, forms a discrete probability distribution on the set {0,1,...,m}, which is known as the "finite discrete arcsine distribution of order m". See Konrad (1969, Section 3.3) and Konrad (1992, Section 2.1, pp. 189-190). (End)

			Triangle T(m,s) (with rows m >= 0 and columns 0 <= s <= m) begins as follows:
    1;
    1,   1;
    3,   2,   3;
   15,   9,   9,  15;
  105,  60,  54,  60, 105;
  945, 525, 450, 450, 525, 945;
  ...
From _Petros Hadjicostas_, May 13 2020: (Start)
With m = 4, we have
J(4) = Integral_{t = 0..Pi/2} (A^2*cos^2(t) + B^2*sin^2(t))^(-4) dt
= Pi/(2^4*A*B*3!) * Sum_{s=0..3} T(3,s)*A^(-2*s)*B(-6+2*s)
= Pi/(96*A*B) * (15*B^(-6) + 9*A^(-2)*B^(-4) + 9*A^(-4)*B^(-2) + 15*A^(-6)). (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167; see a(m,s) (typo in a formula corrected below).

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see p. 85.
Konrad Jacobs, Das kombinatorische Äquivalenzprinzip und das arcsin-Gesetz von E. Sparre Andersen, in: K. Jacobs (eds), Selecta Mathematica I, Heidelberger Taschenbücher, vol 49, Springer, Berlin, Heidelberg, 1969, pp. 53-81; see Lemma 3.3.
Konrad Jacobs, Discrete Stochastics, Springer Basel AG, 1992; see Section 2.1.
Wikipedia, Arcsine distribution.

Central diagonal gives A001757. Other diagonals and columns include A001147, A001193, A001194.

Cf. A000079, A000142, A000165, A000984, A368235.

/* as triangle */ [[Binomial(2*s,s)*Binomial(2*n-2*s, n-s)*Factorial(n)/2^n: s in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 09 2017

A059366 = proc(m, s) option remember; if s = 0  then (2*m)!/(2^m*m!) else
(2*s-1)*(m-s+1)/(s*(2*m-2*s+1)) * A059366(m, s-1) end if; end proc:
seq(print(seq(A059366(m, s), s = 0..m)), m = 0..10) ; # Peter Bala, Apr 14 2024

Table[Binomial[2*s, s]*Binomial[2*n - 2*s, n - s]*n!/2^n, {n, 0, 10}, {s, 0, n}] // Flatten (* G. C. Greubel, Jan 08 2017 *)

for(n=0,10, for(s=0,n, print1(binomial(2*s, s)*binomial(2*n - 2*s, n - s)*n!/2^n, ", "))) \\ G. C. Greubel, Jan 08 2017

T(m+2, s) = (2*m+3)*(T(m+1, s-1) + T(m+1, s)) - 4*(m+1)^2*T(m, s-1).
T(m, s) = m!*Sum_{k=0..s} (-1)^k*2^(2*k-m)*binomial(s, k)*binomial(2*m-2*k, s)*binomial(2*m-2*k-s, m-k). [Typo in Comtet (1974, p. 166) corrected by Petros Hadjicostas, May 12 2020, using Comtet (1967, p. 85).]
From Reinhard Zumkeller, Apr 10 2004: (Start)
T(n,s) = A000984(s)*A000984(n-s)*A000142(n)/A000079(n).
T(n,s) = T(n,n-s).
Sum_{s=0..n} T(n,s) = A000165(n). (End)
From Petros Hadjicostas, May 13 2020: (Start)
T(m,s) = binomial(-1/2, s) * binomial(-1/2, m-s) * (-1)^m * m! * 2^m. [See Konrad (1992, pp. 189-190).]
T(m,m) = A001147(m) = T(m,0) for m >= 0.
T(m,m-1) = A001193(m-1) = T(m,1) for m >= 1.
T(m,m-2) = A001194(m) = T(m,2) for m >= 2.
T(m,m-3) = A001756(m) = T(m,3) for m >= 3.
T(m,floor(m/2)) = A001757(m) = T(m, ceiling(m/2)) for m >= 0.
Lim_{m -> infinity} Sum_{s: s/m <= x} T(m,s)/A000165(m) = (2/Pi)*arcsin(sqrt(x)) for x in [0,1], where the summation is over those s in {0,1,...,m} that satisfy s/m <= x. (End)
From Peter Bala, Apr 14 2024: (Start)
T(m, s) = (2*s - 1)*(m - s + 1)/(s*(2*m - 2*s + 1)) * T(m, s-1) for s >= 1.
T(m, s) = Sum_{i = 0..s} (-1)^(s-i)*binomial(m-i, s-i)*A368235(m, i). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

A002690 a(n) = (n+1) * (2*n)! / n!.

Original entry on oeis.org

1, 4, 36, 480, 8400, 181440, 4656960, 138378240, 4670265600, 176432256000, 7374868300800, 337903056691200, 16838835658444800, 906706535454720000, 52459449551308800000, 3245491278907637760000, 213796737998040637440000, 14940619102451310428160000, 1103945744792235714969600000

Offset: 0

N. J. A. Sloane

Coefficients of orthogonal polynomials.
E.g.f. for series with alternating signs: x/(1+4*x)^(1/2).
Central terms of triangle A245334. - Reinhard Zumkeller, Aug 30 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177.
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177. [Annotated scanned copy]

a(n) = (n+1) * A001813(n) = 2^n * A001193(n+1).
Cf. A002691, A000407.
Cf. A245334.

a002690 n = a245334 (2 * n) n  -- Reinhard Zumkeller, Aug 30 2014

[(n+1) * Factorial(2*n) /Factorial(n): n in [0..20]]; // Vincenzo Librandi, Sep 05 2011

with(combstruct):bin := {B=Union(Z,Prod(B,B))}:
seq (count([B,bin,labeled],size=n+1)*(n+1), n=0..17); # Zerinvary Lajos, Dec 05 2007
A002690 := n -> 2^n*n!*JacobiP(n, -1/2, -n+1, 3):
seq(simplify(A002690(n)), n = 0..18);  # Peter Luschny, Jan 22 2025

Table[((n+1)(2n)!)/n!,{n,0,20}] (* Harvey P. Dale, Sep 04 2011 *)

PARI
```
a(n)=(n+1)*(2*n)!/n!
```

E.g.f.: (1-2*x)/(1-4*x)^(3/2).

a(n) = 2^n*n!*JacobiP(n, -1/2, -n+1, 3). - Peter Luschny, Jan 22 2025

Edited by Ralf Stephan, Mar 21 2004

A167594 A triangle related to the GF(z) formulas of the rows of the ED4 array A167584.

Original entry on oeis.org

1, 2, 2, 9, 2, 13, 60, -12, 68, 76, 525, -300, 774, 132, 789, 5670, -5250, 11820, -3636, 6702, 7734, 72765, -92610, 212415, -143340, 143307, 19086, 110937, 1081080, -1746360, 4286520, -4246200, 4156200, -1204200, 1305000, 1528920

Offset: 1

Johannes W. Meijer, Nov 10 2009

The GF(z) formulas given below correspond to the first ten rows of the ED4 array A167584. The polynomials in their numerators lead to the triangle given above.

			Row 1: GF(z) = 1/(1-z).
Row 2: GF(z) = (2*z + 2)/(1-z)^2.
Row 3: GF(z) = (9*z^2 + 2*z + 13)/(1-z)^3.
Row 4: GF(z) = (60*z^3 - 12*z^2 + 68*z + 76)/(1-z)^4.
Row 5: GF(z) = (525*z^4 - 300*z^3 + 774*z^2 + 132*z + 789)/(1-z)^5.
Row 6: GF(z) = (5670*z^5 - 5250*z^4 + 11820*z^3 - 3636*z^2 + 6702*z + 7734)/(1-z)^6.
Row 7: GF(z) = (72765*z^6 - 92610*z^5 + 212415*z^4 - 143340*z^3 + 143307*z^2 + 19086*z + 110937)/ (1-z)^7.
Row 8: GF(z) = (1081080*z^7 - 1746360*z^6 + 4286520*z^5 - 4246200*z^4 + 4156200*z^3 - 1204200*z^2 + 1305000*z + 1528920)/(1-z)^8.
Row 9: GF(z) = (18243225*z^8 - 35675640*z^7 + 95176620*z^6 -121723560*z^5 + 132769350*z^4 - 73816200*z^3 + 45017100*z^2 + 4887720*z + 28018665) / (1-z)^9.
Row 10: GF(z) = (344594250*z^9 - 790539750*z^8 + 2299457160*z^7 - 3567314520*z^6 + 4441299660*z^5 - 3398138100*z^4 + 2160066600*z^3 - 550619640*z^2 + 421244730*z + 497895210)/(1-z)^10.

A167584 is the ED4 array.
A001193 equals the first left hand column.
A024199 equals the first right hand column.
A002866 equals the row sums.

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).

Original entry on oeis.org

1, 1, -2, 3, -4, 2, 15, -18, 9, -2, 105, -120, 60, -16, 2, 945, -1050, 525, -150, 25, -2, 10395, -11340, 5670, -1680, 315, -36, 2, 135135, -145530, 72765, -22050, 4410, -588, 49, -2, 2027025, -2162160, 1081080, -332640, 69300, -10080, 1008, -64, 2

Offset: 0

Johannes W. Meijer, May 31 2018

The function f(u, p) = (1/2)*(1+1/(sqrt(1-u^2))) * exp(p*sqrt(1-u^2)) was found while studying the Fresnel-Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction, see the Meijer link.
The Taylor expansion of f(u, p) leads to the number triangle T(n, k), see the example section.
Normalization of the triangle terms, dividing the T(n, k) by T(n-k, 0), leads to A084534.
The row sums equal A003436, n >= 2, respectively A231622, n >= 1.

			The first few terms of the Taylor expansion of f(u; p) are:
f(u, p) = exp(p) * (1 + (1-2*p) * u^2/4 + (3-4*p+2*p^2) * u^4/16 + (15-18*p+9*p^2-2*p^3) * u^6/96 + (105-120*p+60*p^2-16*p^3+2*p^4) * u^8/768 + ... )
The first few rows of the T(n, k) triangle are:
n=0:     1
n=1:     1,     -2
n=2:     3,     -4,    2
n=3:    15,    -18,    9,    -2
n=4:   105,   -120,   60,   -16,   2
n=5:   945,  -1050,  525,  -150,  25,  -2
n=6: 10395, -11340, 5670, -1680, 315, -36, 2

J. W. Goodman, Introduction to Fourier Optics, 1996.
A. Papoulis, Systems and Transforms with Applications in Optics, 1968.

Andrew Howroyd, Rows n=0..50 of triangle, flattened
M. J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Optics Communications, Vol. 25, nr. 1, pp. 26-30, 1978.
H. J. Butterweck, General theory of linear, coherent optical data processing systems, Journal of the Optical Society of America, Vol. 67, nr. 1, pp. 60-70, 1977.
J. W. Meijer, A note on optical diffraction, 1979.

Cf. A003436, A231622, A032184, A084534, A127674, A132062, A001497.
Cf. Related to the left hand columns: A001147, A001193, A261065.
Cf. Related to the right hand columns: A280560, A162395, A006011, A040977, A053347, A054334, A266561.

[[n le 0 select 1 else (-1)^k*2^(k-n+1)*Factorial(2*n-k-1)*Binomial(n, k)/Factorial(n-1): k in [0..n]]: n in [1..10]]; // G. C. Greubel, Nov 08 2018

T := proc(n, k): if n=0 then 1 else (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!) fi: end: seq(seq(T(n, k), k=0..n), n=0..8);

Table[If[n==0 && k==0,1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!)], {n, 0, 10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 08 2018 *)

T(n,k) = {if(n==0, 1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 08 2018

T(n, k) = (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*A001147(n-k)*A084534(n, k), n >= 0 and 0 <= k <= n.
T(n, k) = 2^(2*(k-n)+1)*A001147(n-k)*A127674(n, n-k), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*(A001497(n, k) + A132062(n, k)), n >= 1, T(0,0) = 1.

cp's OEIS Frontend

A167584 The ED4 array read by antidiagonals.

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A167591 A triangle related to the a(n) formulas of the rows of the ED4 array A167584.

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A059366 Triangle T(m,s), m >= 0, 0 <= s <= m, arising in the computation of certain integrals.

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A002690 a(n) = (n+1) * (2*n)! / n!.

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A167594 A triangle related to the GF(z) formulas of the rows of the ED4 array A167584.

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A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)*(1+1/(sqrt(1-u^2)))*exp(p*sqrt(1-u^2)).

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A108032 Triangle T(n,k), 0<=k<=n, read by rows, defined by : T(0,0) = 1, T(n,k) = 0 if n

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A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).