A000165 -id:A000165 - cp's OEIS frontend

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

1, 1, 2, 7, 23, 115, 694, 5282, 46066, 456454, 4999004, 59916028, 778525516, 10897964660, 163461964024, 2615361578344, 44460982752488, 800296985768776, 15205638776753680, 304112757426239984, 6386367801916347184

Offset: 1

N. J. A. Sloane

			For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,2413.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. C. Read, personal communication.
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).
Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

N. J. A. Sloane, Table of n, a(n) for n = 1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 9.
Eric Weisstein's World of Mathematics, Rook Number.
Eric Weisstein's World of Mathematics, Rooks Problem.

Cf. A000142, A037223, A037224, A000085, A005635, A099952, A263685.

Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, A000407, A000902, A000900, A000901, A000899, A000903
P:=n->n!; # Gives A000142
G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165
R:=proc(n) local m; if n mod 4 = 2 or n mod 4 = 3 then RETURN(0); fi; m:=floor(n/4); (2*m)!/m!; end; # Gives A122670, A001813
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
B:=proc(n) option remember; if n <= 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(B(n-1)); fi; 2*B(n-2) + (n-2)*B(n-4); end; # Gives A000898 (doubled up)
rho:=n->R(n)/2; # Gives A000407, aerated
beta:=n->B(n)/2; # Gives A000902, doubled up
delta:=n->(D(n)-B(n))/2; # Gives A000900
unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/4; fi; # Gives A000901, doubled up
alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899
unprotect(sigma); sigma:=n-> if n <= 1 then RETURN(1); else P(n)/8+G(n)/8+R(n)/4+D(n)/4; fi; #Gives A000903

c[n_] := Floor[n/2]! 2^Floor[n/2];
r[n_] := If[Mod[n, 4] > 1, 0, m = Floor[n/4]; If[m == 0, 1, (2 m)!/m!]];
d[0] = d[1] = 1; d[n_] := d[n] = (n - 1)d[n - 2] + d[n - 1];
a[1] = 1; a[n_] := (n! + c[n] + 2 r[n] + 2 d[n])/8;
Array[a, 21] (* Jean-François Alcover, Apr 06 2011, after Matthias Engelhardt, further improved by Robert G. Wilson v *)

If n>1 then a(n) = 1/8 * (F(n) + C(n) + 2 * R(n) + 2 * D(n)), where F(n) = A000142(n) [all solutions, i.e., factorials], C(n) = A037223(n) [central symmetric solutions], R(n) = A037224(n) [rotationally symmetric solutions] and D(n) = A000085(n) [symmetric solutions by reflection at a diagonal]. - Matthias Engelhardt, Apr 05 2000

For asymptotics see the Robinson paper.

More terms from David W. Wilson, Jul 13 2003

A039757 Triangle of coefficients in expansion of (x-1)(x-3)(x-5)...(x-(2*n-1)).

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -15, 23, -9, 1, 105, -176, 86, -16, 1, -945, 1689, -950, 230, -25, 1, 10395, -19524, 12139, -3480, 505, -36, 1, -135135, 264207, -177331, 57379, -10045, 973, -49, 1, 2027025, -4098240, 2924172, -1038016, 208054, -24640, 1708, -64, 1, -34459425, 71697105, -53809164, 20570444, -4574934, 626934, -53676, 2796, -81, 1

Offset: 0

Ruedi Suter (suter(AT)math.ethz.ch)

Triangle of B-analogs of Stirling numbers of first kind.

			The triangle T(n, k) begins:
n\k        0        1         2        3        4      5      6    7   8  9
0:         1
1:        -1        1
2:         3       -4         1
3:       -15       23        -9        1
4:       105     -176        86      -16        1
5:      -945     1689      -950      230      -25      1
6:     10395   -19524     12139    -3480      505    -36      1
7:   -135135   264207   -177331    57379   -10045    973    -49    1
8:   2027025 -4098240   2924172 -1038016   208054 -24640   1708  -64   1
9: -34459425 71697105 -53809164 20570444 -4574934 626934 -53676 2796 -81  1
...
row n = 10 :654729075 -1396704420 1094071221 -444647600 107494190 -16486680 1646778 -106800 4335 -100 1
... reformatted and extended. - _Wolfdieter Lang_, May 09 2017

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Priyavrat Deshpande, Krishna Menon, and Anurag Singh, Counting regions of the boxed threshold arrangement, arXiv:2101.12060 [math.CO], 2021.
Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
Z. Kabluchko, V. Vysotsky, and D. Zaporozhets, Convex hulls of random walks, hyperplane arrangements, and Weyl chambers, arXiv preprint arXiv:1510.04073 [math.PR], 2015-2017.

A028338 is unsigned version.
From Johannes W. Meijer, Jun 08 2009: (Start)
A161198 is an unsigned scaled triangle version.
A109692 is an unsigned transposed triangle version.
A000007 equals the row sums. (End)
A000165(n)*(-1)^n (alternating row sums).

nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, 0) := (-1)^n*doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1, m-1)-(2*n-1)*a(n-1, m) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 29 2012

a[n_, m_] := a[n, m] = a[n-1, m-1] - (2*n-1)*a[n-1, m]; a[n_, 0] := (-1)^n*(2*n-1)!!; a[n_, n_] = 1; Table[a[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 16 2012, after Johannes W. Meijer *)

row(n)=Vecrev(prod(i=1,n,'x-2*i+1)) \\ Charles R Greathouse IV, Feb 09 2017

Triangle T(n, k), read by rows, given by [ -1, -2, -3, -4, -5, -6, -7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 20 2005
a(n,m) = a(n-1,m-1) - (2*n-1)*a(n-1,m) with a(n,0) = (-1)^n*(2*n-1)!! and a(n,n) = 1. - Johannes W. Meijer, Jun 08 2009
Exponential Riordan array [1/sqrt(1 + 2*x), 1/2*log(1 + 2*x)] with e.g.f. (1 + 2*x)^((t - 1)/2) = 1 + (t-1)*x + (t-1)*(t-3)*x^2/2! + .... - Peter Bala, Jun 23 2014

A084948 a(n) = Product_{i=0..n-1} (8*i+2).

Original entry on oeis.org

1, 2, 20, 360, 9360, 318240, 13366080, 668304000, 38761632000, 2558267712000, 189311810688000, 15523568476416000, 1397121162877440000, 136917873961989120000, 14513294639970846720000, 1654515588956676526080000, 201850901852714536181760000, 26240617240852889703628800000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

G. C. Greubel, Table of n, a(n) for n = 0..330

Cf. A000079, A000142, A000165, A008544, A001813, A047055, A047657, A048994, A084943, A084947, A084949.

List([0..20], n-> Product([0..n-1], k-> 8*k+2) ); # G. C. Greubel, Aug 18 2019

[1] cat [(&*[8*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

a := n->product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Table[8^n*Pochhammer[1/4, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 8*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(8*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

a(n) = A084943(n)/A000142(n)*A000079(n) = 8^n*Pochhammer(1/4, n) = 1/2*Gamma(n+1/4)*sqrt(2)*Gamma(3/4)*8^n/Pi.
a(n) = (-6)^n*Sum_{k=0..n} (4/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+2)/(2*x*(8*k+2) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 8*x)^(1/4).
a(n) ~ sqrt(2*Pi)*8^n*n^n/(exp(n)*n^(1/4)*Gamma(1/4)). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). - Amiram Eldar, Dec 20 2022

A084949 a(n) = Product_{i=0..n-1} (9*i+2).

Original entry on oeis.org

1, 2, 22, 440, 12760, 484880, 22789360, 1276204160, 82953270400, 6138542009600, 509498986796800, 46873906785305600, 4734264585315865600, 520769104384745216000, 61971523421784680704000, 7932354997988439130112000, 1086732634724416160825344000, 158662964669764759480500224000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Robert Israel, Table of n, a(n) for n = 0..329

Cf. A000165, A008544, A001813, A047055, A047657, A084947, A084948.

Cf. A000079, A000142, A035012, A048994, A084944.

List([0..20], n-> Product([0..n-1], k-> 9*k+2) ); # G. C. Greubel, Aug 19 2019

[1] cat [(&*[9*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 19 2019

a:= n-> product(9*i+2,i=0..n-1); seq(a(j),j=0..20);

Table[9^n*Pochhammer[2/9, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)

vector(20, n, n--; prod(k=0, n-1, 9*k+2)) \\ G. C. Greubel, Aug 19 2019

[product(9*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

a(n) = A084944(n)/A000142(n)*A000079(n) = 9^n*Pochhammer(2/9, n) = 9^n*Gamma(n+2/9)/Gamma(2/9).
a(n) = (-7)^n*Sum_{k=0..n} (9/7)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
E.g.f.: (1-9*x)^(-2/9). - Robert Israel, Mar 22 2017
D-finite with recurrence: a(n) + (-9*n+7)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/9^7)^(1/9)*(Gamma(2/9) - Gamma(2/9, 1/9)). - Amiram Eldar, Dec 21 2022
a(n) ~ sqrt(2*Pi) * (9/e)^n * n^(n-5/18) / Gamma(2/9). - Amiram Eldar, Aug 30 2025

A129890 a(n) = (2n+2)!! - (2n+1)!!.

Original entry on oeis.org

1, 5, 33, 279, 2895, 35685, 509985, 8294895, 151335135, 3061162125, 68000295825, 1645756410375, 43105900812975, 1214871076343925, 36659590336994625, 1179297174137457375, 40288002704636061375, 1456700757237661060125

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 04 2007

Previous name was: Difference between the double factorial of the n-th nonnegative even number and the double factorial of the n-th nonnegative odd number.
In other words, a(n) = b(2n+2)-b(2n+1), where b = A006882. - N. J. A. Sloane, Dec 14 2011 [Corrected Peter Luschny, Dec 01 2014]
a(n) is the number of linear chord diagrams on 2n+2 vertices with one marked chord such that none of the remaining n chords are contained within the marked chord, see [Young]. - Donovan Young, Aug 11 2020

			2!! - 1!! =  2 -  1 =  1;
4!! - 3!! =  8 -  3 =  5;
6!! - 5!! = 48 - 15 = 33.

Selden Crary, Richard Diehl Martinez, and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
N. Ochiumi, On the total sum of number of nodes covering a given number of leaves in an unordered binary tree
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A006882, A122649, A202212, A336599.

seq(doublefactorial(2*n+2)-doublefactorial(2*n+1),n=0..9); # Peter Luschny, Dec 01 2014

a[n_] := (2n+2)!! - (2n+1)!!;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 30 2018 *)

E.g.f.: 2/((1-2*x)^2)-1/[(1-2*x)*sqrt(1-2*x)]. - Sergei N. Gladkovskii, Dec 04 2011
a(n) = (2*n+1)*a(n-1) + A000165(n). - Philippe Deléham, Oct 28 2013
Conjecture: a(n) = (2*n + 2)*(2*n + 2)! * Sum_{k >= 1} (-1)^(k+1)/Product_{j = 0..n+1} (k + 2*j). - Peter Bala, Jul 06 2025

New name from Peter Luschny, Dec 01 2014

A051579 a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 7, 63, 693, 9009, 135135, 2297295, 43648605, 916620705, 21082276215, 527056905375, 14230536445125, 412685556908625, 12793252264167375, 422177324717523375, 14776206365113318125, 546719635509192770625

Offset: 0

Wolfdieter Lang

Row m=5 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A051577, A051578 (rows m=0..4).

Cf. A051580, A051581, A051582, A051583, A178647.

List([0..20], n-> Product([0..n-1], j-> 2*j+7) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+7: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

df:=doublefactorial; seq(df(2*n+5)/df(5), n = 0..20); # G. C. Greubel, Nov 12 2019

Table[2^n*Pochhammer[7/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=1,n-1, 2*j+5) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+7) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+5)!!/4!!.
E.g.f.: 1/(1-2*x)^(7/2).
a(n) ~ 8/15*sqrt(2)*n^3*2^n*e^-n*n^n*(1 + 107/24*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: G(0)/(10*x) -1/(5*x), where G(k)= 1 + 1/(1 - x*(2*k+5)/(x*(2*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 13 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 7)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 7*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 7*x/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - ... - (2*n + 5)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 7*x/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 15 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 20, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 15 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 10, where erfi is the imaginary error function. (End)

A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang Sep 14 2007

This is a Jabotinsky type exponential convolution triangle related to A001147 (double factorials). For Jabotinsky type triangles See the D. E. Knuth reference given under A039692.
The subtriangle (n>=m>=1) is A001497(n,m) (Bessel).
For the combinatorial interpretation in terms of unordered forests of increasing plane trees see the W. Lang comment and example under A001497.
This is a special type of Sheffer triangle. See the S. Roman reference given under A048854 (the notation here differs).
This triangle (or the A001497 subtriangle) appears as generalized Stirling numbers of the second kind, S2p(-1,n,m):=S2(-k;m,m)*(-1)^(n-m) for k=1, eqs. (27)-(29) of the W. Lang reference.
Also the Bell transform of the double factorial of odd numbers A001147. For the Bell transform of the double factorial of even numbers A000165 see A039683. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015

			[1]
[0,      1]
[0,      1,      1]
[0,      3,      3,     1]
[0,     15,     15,     6,     1]
[0,    105,    105,    45,    10,    1]
[0,    945,    945,   420,   105,   15,   1]
[0,  10395,  10395,  4725,  1260,  210,  21,  1]
[0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]

Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012
Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78) [Emanuele Munarini, Oct 10 2017]

Leonard Carlitz, A Note on the Bessel Polynomials, Duke Math. J. 24 (2) (1957), 151-162. [_Emanuele Munarini_, Oct 10 2017]
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.

Columns m=1: A001147.
Row sums give [1, A001515]. Alternating row sums give [1, -A000806].
Cf. A122850. - R. J. Mathar, Mar 20 2009
Cf. A039683, A264428.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
# Alternative:
egf := exp(y*(1 - sqrt(1 - 2*x))): serx := series(egf, x, 12):
coefx := n -> n!*coeff(serx, x, n): row := n -> seq(coeff(coefx(n), y, k), k = 0..n): for n from 0 to 8 do row(n) od;  # Peter Luschny, Apr 25 2024

Table[If[k <= n, Binomial[2n-2k,n-k] Binomial[2n-k-1,k-1] (n-k)!/2^(n-k), 0], {n, 0, 6}, {k, 0, n}] // Flatten (* Emanuele Munarini, Oct 10 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(2#-1)!!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

# uses[bell_transform from A264428]
def A132062_row(n):
    a = sloane.A001147
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A132062_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

a(n,m)=0 if n

E.g.f. m-th column ((x*f2p(1;x))^m)/m!, m>=0. with f2p(1;x):=1-sqrt(1-2*x)= x*c(x/2) with the o.g.f.of A000108 (Catalan).

From Emanuele Munarini, Oct 10 2017: (Start)

a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).

The row polynomials p_n(x) (studied by Carlitz) satisfy the recurrence: p_{n+2}(x) - (2*n+1)*p_{n+1}(x) - x^2*p_n(x) = 0. (End)

T(n, k) = n! [y^k] [x^n] exp(y*(1 - sqrt(1 - 2*x))). - Peter Luschny, Apr 25 2024

A185896 Triangle of coefficients of (1/sec^2(x))*D^n(sec^2(x)) in powers of t = tan(x), where D = d/dx.

Original entry on oeis.org

1, 0, 2, 2, 0, 6, 0, 16, 0, 24, 16, 0, 120, 0, 120, 0, 272, 0, 960, 0, 720, 272, 0, 3696, 0, 8400, 0, 5040, 0, 7936, 0, 48384, 0, 80640, 0, 40320, 7936, 0, 168960, 0, 645120, 0, 846720, 0, 362880, 0, 353792, 0, 3256320, 0, 8951040, 0, 9676800, 0, 3628800

Offset: 0

Peter Bala, Feb 07 2011

DEFINITION
Define polynomials R(n,t) with t = tan(x) by
... (d/dx)^n sec^2(x) = R(n,tan(x))*sec^2(x).
The first few are
... R(0,t) = 1
... R(1,t) = 2*t
... R(2,t) = 2 + 6*t^2
... R(3,t) = 16*t + 24*t^3.
This triangle shows the coefficients of R(n,t) in ascending powers of t called the tangent number triangle in [Hodges and Sukumar].
The polynomials R(n,t) form a companion polynomial sequence to Hoffman's two polynomial sequences - P(n,t) (A155100), the derivative polynomials of the tangent and Q(n,t) (A104035), the derivative polynomials of the secant. See also A008293 and A008294.
COMBINATORIAL INTERPRETATION
A combinatorial interpretation for the polynomial R(n,t) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges].
A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}. They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube.
Let x_1,...,x_n be a signed permutation.
Then 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0.
For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0).
Let sc be the number of sign changes through a snake
... sc = #{i, 1 <= i <= n-1, x_i*x_(i+1) < 0}.
For example, the snake 0 4 -3 -1 -2 0  has sc = 1. The polynomial R(n,t) is the generating function for the sign change statistic on snakes of type S(n+1;0,0):
... R(n,t) = sum {snakes in S(n+1;0,0)} t^sc.
See the example section below for the cases n=1 and n=2.
PRODUCTION MATRIX
Define three arrays R, L, and S as
... R = superdiag[2,3,4,...]
... L = subdiag[1,2,3,...]
... S = diag[2,4,6,...]
with the indicated sequences on the main superdiagonal, the main subdiagonal and main diagonal, respectively, and 0's elsewhere. The array R+L is the production array for this triangle: the first row of (R+L)^n produces the n-th row of the triangle.
On the vector space of complex polynomials the array R, the raising operator, represents the operator p(x) - > d/dx (x^2*p(x)), and the array L, the lowering operator, represents the differential operator d/dx - see Formula (4) below.
The three arrays satisfy the commutation relations
... [R,L] = S, [R,S] = 2*R, [L,S] = -2*L
and hence give a representation of the Lie algebra sl(2).

			Table begins
  n\k|.....0.....1.....2.....3.....4.....5.....6
  ==============================================
  0..|.....1
  1..|.....0.....2
  2..|.....2.....0.....6
  3..|.....0....16.....0....24
  4..|....16.....0...120.....0...120
  5..|.....0...272.....0...960.....0...720
  6..|...272.....0..3696.....0..8400.....0..5040
Examples of recurrence relation
  T(4,2) = 3*(T(3,1) + T(3,3)) = 3*(16 + 24) = 120;
  T(6,4) = 5*(T(5,3) + T(5,5)) = 5*(960 + 720) = 8400.
Example of integral formula (6)
... Integral_{t = -1..1} (1-t^2)*(16-120*t^2+120*t^4)*(272-3696*t^2+8400*t^4-5040*t^6) dt = 2830336/1365 = -2^13*Bernoulli(12).
Examples of sign change statistic sc on snakes of type (0,0)
= = = = = = = = = = = = = = = = = = = = = =
.....Snakes....# sign changes sc.......t^sc
= = = = = = = = = = = = = = = = = = = = = =
n=1
...0 1 -2 0...........1................t
...0 2 -1 0...........1................t
yields R(1,t) = 2*t;
n=2
...0 1 -2 3 0.........2................t^2
...0 1 -3 2 0.........2................t^2
...0 2 1 3 0..........0................1
...0 2 -1 3 0.........2................t^2
...0 2 -3 1 0.........2................t^2
...0 3 1 2 0..........0................1
...0 3 -1 2 0.........2................t^2
...0 3 -2 1 0.........2................t^2
yields
R(2,t) = 2 + 6*t^2.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
K. Boyadzhiev, Derivative Polynomials for tanh, tan, sech and sec in Explicit Form, arXiv:0903.0117 [math.CA], 2009-2010.
M-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A (2007) 463, 2401-2414 doi:10.1098/rspa.2007.0001
Michael E. Hoffman, Derivative polynomials, Euler polynomials,and associated integer sequences, Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R21.
Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.
M. Josuat-Verges, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929 [math.CO], 2010.
Shi-Mei Ma, Qi Fang, Toufik Mansour, Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021

Cf. A000182, A000364, A000828 (row sums), A008292, A008293, A008294, A104035, A155100.

R = proc(n) option remember;
if n=0 then RETURN(1);
else RETURN(expand(diff((u^2+1)*R(n-1), u))); fi;
end proc;
for n from 0 to 12 do
t1 := series(R(n), u, 20);
lprint(seriestolist(t1));
od:

Table[(-1)^(n + 1)*(-1)^((n - k)/2)*Sum[j!*StirlingS2[n + 1, j]*2^(n + 1 - j)*(-1)^(n + j - k)*Binomial[j - 1, k], {j, k + 1, n + 1}], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

{T(n, k) = if( n<0 || k<0 || k>n, 0, if(n==k, n!, (k+1)*(T(n-1, k-1) + T(n-1, k+1))))};

{T(n, k) = my(A); if( n<0 || k>n, 0, A=1; for(i=1, n, A = ((1 + x^2) * A)'); polcoeff(A, k))}; /* Michael Somos, Jun 24 2017 */

GENERATING FUNCTION
E.g.f.:
(1)... F(t,z) = 1/(cos(z)-t*sin(z))^2 = Sum_{n>=0} R(n,t)*z^n/n! = 1 + (2*t)*z + (2+6*t^2)*z^2/2! + (16*t+24*t^3)*z^3/3! + ....
The e.g.f. equals the square of the e.g.f. of A104035.
Continued fraction representation for the o.g.f:
(2)... F(t,z) = 1/(1-2*t*z - 2*(1+t^2)*z^2/(1-4*t*z -...- n*(n+1)*(1+t^2)*z^2/(1-2*n*(n+1)*t*z -....
RECURRENCE RELATION
(3)... T(n,k) = (k+1)*(T(n-1,k-1) + T(n-1,k+1)).
ROW POLYNOMIALS
The polynomials R(n,t) satisfy the recurrence relation
(4)... R(n+1,t) = d/dt{(1+t^2)*R(n,t)} with R(0,t) = 1.
Let D be the derivative operator d/dt and U = t, the shift operator.
(5)... R(n,t) = (D + DUU)^n 1
RELATION WITH OTHER SEQUENCES
A) Derivative Polynomials A155100
The polynomials (1+t^2)*R(n,t) are the polynomials P_(n+2)(t) of A155100.
B) Bernoulli Numbers A000367 and A002445
Put S(n,t) = R(n,i*t), where i = sqrt(-1). We have the definite integral evaluation
(6)... Integral_{t = -1..1} (1-t^2)*S(m,t)*S(n,t) dt = (-1)^((m-n)/2)*2^(m+n+3)*Bernoulli(m+n+2).
The case m = n is equivalent to the result of [Grosset and Veselov]. The methods used there extend to the general case.
C) Zigzag Numbers A000111
(7)... R_n(1) = A000828(n+1) = 2^n*A000111(n+1).
D) Eulerian Numbers A008292
The polynomials R(n,t) are related to the Eulerian polynomials A(n,t) via
(8)... R(n,t) = (t+i)^n*A(n+1,(t-i)/(t+i))
with the inverse identity
(9)... A(n+1,t) = (-i/2)^n*(1-t)^n*R(n,i*(1+t)/(1-t)),
where {A(n,t)}n>=1 = [1,1+t,1+4*t+t^2,1+11*t+11*t^2+t^3,...] is the sequence of Eulerian polynomials and i = sqrt(-1).
E) Ordered set partitions A019538
(10)... R(n,t) = (-2*i)^n*T(n+1,x)/x,
where x = i/2*t - 1/2 and T(n,x) is the n-th row po1ynomial of A019538;
F) Miscellaneous
Column 1 is the sequence of tangent numbers - see A000182.
A000670(n+1) = (-i/2)^n*R(n,3*i).
A004123(n+2) = 2*(-i/2)^n*R(n,5*i).
A080795(n+1) =(-1)^n*(sqrt(-2))^n*R(n,sqrt(-2)). - Peter Bala, Aug 26 2011
From Leonid Bedratyuk, Aug 12 2012: (Start)
T(n,k) = (-1)^(n+1)*(-1)^((n-k)/2)*Sum_{j=k+1..n+1} j! *stirling2(n+1,j) *2^(n+1-j) *(-1)^(n+j-k) *binomial(j-1,k), see A059419.
Sum_{j=i+1..n+1}((1-(-1)^(j-i))/(2*(j-i))*(-1)^((n-j)/2)*T(n,j))=(n+1)*(-1)^((n-1-i)/2)*T(n-1,i), for n>1 and  0G.f.:  1/G(0,t,x), where G(k,t,x) = 1 - 2*t*x - 2*k*t*x - (1+t^2)*(k+2)*(k+1)*x^2/G(k+1,t,x); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Dec 27 2013

A217626 First differences of A215940, or first differences of permutations of (0,1,2,...,m-1) reading them as decimal numbers, divided by 9 (with 10>=m, and m! > n).

Original entry on oeis.org

1, 9, 2, 9, 1, 78, 1, 19, 3, 8, 2, 77, 2, 8, 3, 19, 1, 78, 1, 9, 2, 9, 1, 657, 1, 9, 2, 9, 1, 178, 1, 29, 4, 7, 3, 66, 2, 18, 4, 18, 2, 67, 1, 19, 3, 8, 2, 646, 1, 19, 3, 8, 2, 67, 1, 29, 4, 7, 3, 176, 3, 7, 4, 29, 1, 67, 2, 8, 3, 19, 1, 646, 2, 8, 3, 19, 1

Offset: 1

R. J. Cano, Oct 04 2012

Terms do not depend on the choice of m, provided that m!>n (the index of the considered term), and the numbers associated to a permutation s of {0,...,m-1} are N(s) = Sum_{i=1..m} s(i)*10^(m-i). This defines the present sequence for any arbitrarily large index, not limited to n <= 10!, for example.
Similar sequences might be built in another base b, they would always start (1, b-1, 2, b-1, 1, ...). The partial sums of this kind of sequence would yield the analog of A215940 in the corresponding base.
There are at least two palindromic patterns which are repeated throughout this sequence: one of them is "1,b-1,2,b-1,1" (It is optional here whether or not to include the 1's), another is built from the first 4!-1 terms (See the corresponding link for details).
Also, for 1<=n<=(9!)-1: The repeating parts in the first differences of A030299 divided by nine, i.e. a(n) = A219664(n)/9. - Antti Karttunen, Dec 18 2012. Edited by: R. J. Cano, May 09 2017
There are more palindromic patterns than those mentioned above: Similar to the first 3!-1 and the first 4!-1 terms, the first k!-1 terms are repeated for all other k>4. Frequent are also multiples of these, e.g., k*[1,9,2,9,1] = [2,18,4,18,2], [3,27,6,27,3], ...), [1, 19, 3, 8, 2, 67, 1, 29, 4, 7, 3, 176, 3, 7, 4, 29, 1, 67, 2, 8, 3, 19, 1], and others. The "middle part" of roughly half the length (e.g., [9,2,9] or [67,...,67] in the last example), is repeated even more frequently. - M. F. Hasler, Jan 14 2013
From R. J. Cano, Apr 04 2016: (Start)
Conjecture 1: Given 1A217626 and so on).
Lemma: Let P be an arbitrary set consisting of m integers; let x[i] be an element in P (with 1<=i<=m); let y[j] = x[j+1] - x[j] (with 1 <= j <= m-1) be the 1st differences of P. These differences are symmetric if y[j]=y[m-j] which for P implies the condition x[j]+x[m-j+1]=x[j+1]+x[m-j];
Consequence: When m=n! and P is a set with all the permutations for the letters 0..n-1, the preceding lemma implies P has associated at least a set Q such that 1st differences in Q are symmetric.
Generating algorithm: Such Q can be built based upon P and the condition given by the preceding lemma if it is removed from P (until P becomes empty) its 1st element tau, inserting them both in Q tau and its arithmetic complement to repdigit (n-1)*111...1 (n times 1) removing the mentioned complement from P.
Conjecture 2: The autosimilarity shown by a(n) is a consequence of the fact that the corresponding P is the set of the n! permutations in increasing sequence for the letters 0..n-1, and Q=P (it holds if they are replaced "a(n)" and "increasing" respectively with "-1*a(n)" and "decreasing").
Note: "Q=P" is a necessary but not sufficient condition for observing the autosimilarity in a(n).
Application: The "generating algorithm" described previously might be potentially useful for parallel computing. In combination with the partition scheme proposed at links in A237265, and multiple indirection. For example notice that in such sense an algorithm for generating k! permutations with an increasing sequence would require only k!/2 iterations because the other half would be already determined by symmetry.
Conjecture 3: For n>2, given P the set of permutations in increasing sequence for the letters 0..n-1, there are distributed with a symmetric pattern among its (n!)! permutations all those A000165(n!\2) of them such that their 1st differences are symmetric. Moreover by setting to zero the other elements whose 1st differences are not symmetric, we obtain an antisymmetric sequence.
(End)
Conjecture 4: If 2<=mR. J. Cano, Apr 19 2017
Consider the first y!-1 terms for even y; The central term a(y!/2) is determined by the difference between the (y/2+1)th row from the y-th matrix defining the irregular table in A237265 and the consecutive permutation preceding it in lexicographic order (See EXAMPLE). - R. J. Cano, May 09 2017

			a(1)= A215940(2) - A215940(1) = 1 - 0 = 1.
a(2)= A215940(3) - A215940(2) = 10-01 = 9.
a(3)= A215940(4) - A215940(3) = 12-10 = 2.
a(4)= A215940(5) - A215940(4) = 21-12 = 9.
a(5)= A215940(6) - A215940(5) = 22-21 = 1.
From _R. J. Cano_, May 09 2017: (Start)
On the central terms for subsequences consisting of the first y!-1 terms with even y: Let us pick y=4; The first y!-1=23 terms are: (1,9,2,9,1,78,1,19,3,8,2,77,2,8,3,19,1,78,1,9,2,9,1) the central term there is a(12)=77.
If we look into A237265, the 4th matrix defining it contains as its (4/2+1)th or third row, the permutation 3124 which in lexicographic order is preceded by 2431, therefore by subtracting and dividing by 9 we obtain: (3124-2431)/9 = 693/9 = (2013-1320)/9 = 77 = a(12). (End)

A. Karttunen, Table of n, a(n) for n = 1..40319
R. J. Cano, Compressed file: a(n) for n = 1..3628799, the first 10!-1 terms of this sequence.
R. J. Cano, Additional information about this sequence.
R. J. Cano, Template for a base-independent sequencer in C.

Cf. A215940, A207324, A219664, A220664, A030299.

Cf. A219995 [ On the summation of 1/a(n) ].

C
```
// See LINKS.
```
Maple
```
A217626:=n->A215940(n+1)-A215940(n);
```

maxm = 5; Table[dd = FromDigits /@ Permutations[Range[m]]; (Drop[dd, If[m == 1, 0, (m - 1)!]] - First[dd])/9, {m, 1, maxm}] // Flatten // Differences (* Jean-François Alcover, Apr 25 2013 *)

first_terms(n)={n=max(3,n);my(m:small=n!);my(a:vec=vector(m-1),i:small=0,x:vec=numtoperm(n,0),y:vec,z:vec,u:small,B:small=11);m\=2;m--;while(i++<=m,u=!(i%6);y=numtoperm(n,i);z=(y-x)[1..n-1];if(u,z=vector(#z,j,vecsum(z[1..j])));a[i]=fromdigits(z,B-u);a[#a-i+1]=a[i];x=y;);z=(numtoperm(n,m+1)-y)[1..n-1];a[m+1]=fromdigits(vector(#z,j,vecsum(z[1..j])),B--);return(a)} \\ Computes the first either 5 or n!-1 terms. - R. J. Cano, May 28 2017

(define (A217626 n) (/ (A219664 n) 9)) ;; - Antti Karttunen, Dec 18 2012

a(n) = A215940(n+1) - A215940(n).
a(n) = A219664(n)/9, for n=1..362879. - Antti Karttunen, Dec 18 2012
a(n) = A209280(n)/9, for n < 9!. - M. F. Hasler, Jan 12 2013

Definition simplified by M. F. Hasler, Jan 12 2013

A334380 Decimal expansion of Sum_{k>=0} (-1)^k/((2*k)!!)^2.

Original entry on oeis.org

7, 6, 5, 1, 9, 7, 6, 8, 6, 5, 5, 7, 9, 6, 6, 5, 5, 1, 4, 4, 9, 7, 1, 7, 5, 2, 6, 1, 0, 2, 6, 6, 3, 2, 2, 0, 9, 0, 9, 2, 7, 4, 2, 8, 9, 7, 5, 5, 3, 2, 5, 2, 4, 1, 8, 6, 1, 5, 4, 7, 5, 4, 9, 1, 1, 9, 2, 7, 8, 9, 1, 2, 2, 1, 5, 2, 7, 2, 4, 4, 0, 1, 6, 7, 1, 8, 0, 6, 0, 0, 0, 9, 8, 9, 1, 5, 6, 3, 3, 9, 7, 4, 9, 2, 9, 2, 5, 9, 8, 2

Offset: 0

Ilya Gutkovskiy, Apr 25 2020

This constant is transcendental.

			1/(4^0*0!^2) - 1/(4^1*1!^2) + 1/(4^2*2!^2) - 1/(4^3*3!^2) + ... = 0.765197686557966551449717526...

Cf. A000165, A002454.

Bessel function values: this sequence (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselJ[0, 1], 10, 110] [[1]]

besselj(0, 1) \\ Michel Marcus, Apr 26 2020

Equals BesselJ(0,1).

Equals BesselI(0,i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

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A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

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A039757 Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).

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A084948 a(n) = Product_{i=0..n-1} (8*i+2).

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A084949 a(n) = Product_{i=0..n-1} (9*i+2).

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A129890 a(n) = (2*n+2)!! - (2*n+1)!!.

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A051579 a(n) = (2*n+5)!!/5!!, related to A001147 (odd double factorials).

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A039757 Triangle of coefficients in expansion of (x-1)(x-3)(x-5)...(x-(2*n-1)).

A129890 a(n) = (2n+2)!! - (2n+1)!!.