A072374 -id:A072374 - cp's OEIS frontend

A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n.

1, 1, 3, 8, 31, 147, 853, 5824, 45741, 405845, 4012711, 43733976, 520795003, 6726601063, 93651619881, 1398047697152, 22275111534553, 377278848390249, 6768744159489931, 128228860181918440, 2557808459478878871, 53585748788874537851, 1176328664895760953853

Offset: 0

N. J. A. Sloane and John Riordan

Row sums of A086764. - Philippe Deléham, Apr 27 2004

a(n+2m) == a(n) (mod m) for all n and m. - Robert Israel, Dec 06 2016

			G.f. = 1 + x + 3*x^2 + 8*x^3 + 31*x^4 + 147*x^5 + 853*x^6 + 5824*x^7 + ...

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

T. D. Noe, Table of n, a(n) for n = 0..100
John Riordan and N. J. A. Sloane, Correspondence, 1974

Cf. A008279, A072374, A086764, A143409.

f:= gfun:-rectoproc({a(n) -(n-1)*a(n-1)-(n-2)*a(n-2)+a(n-3)-2=0,a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 06 2016

t = {1, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]] + 1 + (-1)^n], {n, 2, 20}] (* T. D. Noe, Oct 07 2013 *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n},{},-1];
Table[Sum[(-1)^k T[n,k], {k,0,n}], {n,0,22}] (* Peter Luschny, Oct 05 2017 *)

Diagonal sums of reverse of permutation triangle A008279. a(n) = Sum_{k=0..floor(n/2)} (n-k)!/k!. - Paul Barry, May 12 2004
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(n-2k)!. - Paul Barry Dec 15 2010
G.f.: 1/(1-x^2-x/(1-x/(1-x^2-2x/(1-2x/(1-x^2-3x/(1-3x/(1-x^2-4x/(1-4x/(1-.... (continued fraction);
G.f.: 1/(1-x-x^2-x^2/(1-3x-x^2-4x^2/(1-5x-x^2-9x^2/(1-7x-x^2-16x^2/(1-... (continued fraction). - Paul Barry, Dec 15 2010
G.f.: hypergeom([1,1],[],x/(1-x^2))/(1-x^2). - Mark van Hoeij, Nov 08 2011
G.f.: 1/Q(0), where Q(k)= 1 - x^2 - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013
From Robert Israel, Dec 06 2016: (Start)
a(2m) = hypergeom([1,-m,m+1],[],-1).
a(2m+1) = hypergeom([1,-m,m+2],[],-1)*(m+1).
a(2m-1) + a(2m+1) = (2m+1) a(2m). (End)
0 = a(n)*(-a(n+2) - a(n+3)) + a(n+1)*(-2 + a(n+1) - 2*a(n+3) + a(n+4)) + a(n+2)*(-2*a(n+3) + a(n+4)) + a(n+3)*(+2 - a(n+3)) if n >= 0. - Michael Somos, Dec 06 2016
0 = a(n)*(-a(n+2) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - a(n+3) + 3*a(n+4) - a(n+5)) + a(n+2)*(-a(n+3) + a(n+4)) + a(n+3)*(-a(n+4) + a(n+5)) + a(n+4)*(-a(n+4)) if n >= 0. - Michael Somos, Dec 06 2016
a(n) = Sum_{k=0..n} (-1)^k*hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017
D-finite with recurrence: a(n) -n*a(n-1) +(n-2)*a(n-3) -a(n-4)=0. - R. J. Mathar, Apr 29 2020
a(n) ~ n! * (1 + 1/n + 1/(2*n^2) + 2/(3*n^3) + 25/(24*n^4) + 77/(40*n^5) + 2971/(720*n^6) + 6287/(630*n^7) + 1074809/(40320*n^8) + 28160749/(362880*n^9) + ...). - Vaclav Kotesovec, Nov 25 2022

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 25 2004

A122852 Row sums of number triangle A122851.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 24, 51, 122, 291, 756, 1979, 5526, 15627, 46496, 140451, 442194, 1414931, 4687212, 15785451, 54764846, 193129659, 698978136, 2570480147, 9672977706, 36967490691, 144232455524, 571177352091, 2304843053382, 9434493132011, 39289892366736

Offset: 0

Paul Barry, Sep 14 2006

Essentially the same as A072374. - R. J. Mathar, Jun 18 2008

Diagonal sums of A008279. - Paul Barry, Feb 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jonathan Fang, Zachary Hamaker, and Justin Troyka, On pattern avoidance in matchings and involutions, arXiv:2009.00079 [math.CO], 2020. See Theorem 1.6 (b).
Guo-Niu Han, Hankel Continued fractions and Hankel determinants of the Euler numbers, arXiv:1906.00103 [math.CO], 2019. See p. 27.
Qiong Qiong Pan and Jiang Zeng, The gamma-coefficients of Branden's (p,q)-Eulerian polynomials and André permutations, arXiv:1910.01747 [math.CO], 2019.

Cf. A072374, A008279.

Cf. A357532, A357533, A357570.

Table[Sum[Binomial[n-k,k]*k!,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Feb 08 2014 *)

a(n) = sum(k=0, n, binomial(k,n-k)*(n-k)!); \\ Michel Marcus, Sep 02 2020

a(n) = Sum{k=0..n} C(k,n-k)*(n-k)!.
From Paul Barry, Feb 11 2009: (Start)
G.f.: 1/(1-x-x^2/(1-x^2/(1-x-2x^2/(1-2x^2/(1-x-3x^2/(1-3x^2/(1-x-4x^2/(1-4x^2/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*k!. (End)
D-finite with recurrence -2*a(n) + 3*a(n-1) + (n-1)*a(n-2) + (-n+1)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012. Proof in [Han 2019]
a(n) ~ sqrt(Pi) * exp(sqrt(n/2) - n/2 + 1/8) * n^((n+1)/2) / 2^(n/2+1) * (1 + 37/(48*sqrt(2*n))). - Vaclav Kotesovec, Feb 08 2014
a(n) = (a(n-1) + n * a(n-2) + 1)/2 for n > 1. - Seiichi Manyama, Nov 19 2022

More terms from Vaclav Kotesovec, Jun 04 2019

A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2k)!/(n-3k)!.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 211, 414, 799, 1588, 3267, 6706, 13999, 30024, 64723, 141142, 314271, 705724, 1599619, 3685338, 8573167, 20112016, 47804499, 114743614, 277615903, 679057092, 1676636611, 4171532674, 10477002159, 26545428568, 67755344467, 174386589606

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A072374, A122852, A357533, A357570.

a(n) = sum(k=0, n\3, (n-2*k)!/(n-3*k)!);

a(n) = (2 * a(n-1) + n * a(n-3) + 1)/3 for n > 2.

a(n) ~ c * n^(n/3 + 1/2) / (3^(n/3) * exp(n/3 - n^(2/3)/3^(2/3) - 2*n^(1/3) / 3^(7/3))) * (1 + 1235/(729 * 3^(2/3) * n^(1/3)) + 9452027/(15943230 * 3^(1/3) * n^(2/3)) + 16015315669/(41841412812*n)), where c = 0.50682110703119..., conjecture: c = exp(4/81) * sqrt(2*Pi) / 3^(3/2). - Vaclav Kotesovec, Nov 25 2022

A113287 Triangle T, read by rows, where row n of T equals row n of matrix (n+1)-th power of triangle A112555.

Original entry on oeis.org

1, 2, 1, -3, 0, 1, 4, 4, 4, 1, -5, -10, -10, 0, 1, 6, 18, 24, 12, 6, 1, -7, -28, -49, -42, -21, 0, 1, 8, 40, 88, 104, 72, 24, 8, 1, -9, -54, -144, -216, -198, -108, -36, 0, 1, 10, 70, 220, 400, 460, 340, 160, 40, 10, 1, -11, -88, -319, -682, -946, -880, -550, -220, -55, 0, 1

Offset: 0

Paul D. Hanna, Oct 23 2005

Remarkably, the matrix logarithm (A113290) is an integer triangle. Matrix m-th power of A112555 = I + m*(A112555 - I) where I = identity matrix.

			Triangle begins:
1;
2,1;
-3,0,1;
4,4,4,1;
-5,-10,-10,0,1;
6,18,24,12,6,1;
-7,-28,-49,-42,-21,0,1;
8,40,88,104,72,24,8,1;
-9,-54,-144,-216,-198,-108,-36,0,1;
10,70,220,400,460,340,160,40,10,1; ...

Cf. A112555, A113288 (inverse), A113290 (log), A113291, A072374.

{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff(1/(1-x*y)+x*(x+2)/((1-x*y)^2*(1+x+x*y)^2),n,X),k,Y)}

G.f.: A(x, y) = 1/(1-x*y) + x*(x+2)/((1-x*y)^2*(1+x+x*y)^2).

A113290 Matrix logarithm of triangle A113287.

Original entry on oeis.org

0, 2, 0, -3, 0, 0, 6, 4, 4, 0, -10, -10, -10, 0, 0, 19, 24, 30, 12, 6, 0, -35, -49, -70, -42, -21, 0, 0, 72, 104, 164, 128, 84, 24, 8, 0, -150, -216, -360, -324, -252, -108, -36, 0, 0, 343, 480, 820, 800, 710, 400, 180, 40, 10, 0, -803, -1089, -1870, -1892, -1826, -1210, -660, -220, -55, 0, 0

Offset: 0

Paul D. Hanna, Oct 23 2005

			Triangle begins:
0;
2,0;
-3,0,0;
6,4,4,0;
-10,-10,-10,0,0;
19,24,30,12,6,0;
-35,-49,-70,-42,-21,0,0;
72,104,164,128,84,24,8,0;
-150,-216,-360,-324,-252,-108,-36,0,0;
343,480,820,800,710,400,180,40,10,0; ...

Cf. A113287, A113288, A113291 (column 1), A113292 (column 0), A072374.

{T(n,k)=local(x=X+O(X^(n+2)),y=Y+O(Y^(n+2)),M=matrix(n+1,n+1,r,c,if(r==c,1, if(r>c,r*polcoeff(polcoeff(1/(1-x*y)+x/((1-x*y)*(1+x+x*y)),r-1,X),c-1,Y))))); if(n

T(n, 1) = (n+1)*A113291(n).

A357533 a(n) = Sum_{k=0..floor(n/4)} (n-3k)!/(n-4k)!.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 46, 77, 128, 205, 338, 581, 1012, 1733, 2990, 5293, 9536, 17117, 30778, 56165, 104108, 193621, 360662, 677693, 1289080, 2467373, 4735826, 9142837, 17814308, 34950245, 68835118, 136197581, 271384112, 544302973, 1096578410, 2218459013, 4513377436

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A072374, A122852, A357532, A357570.

a(n) = sum(k=0, n\4, (n-3*k)!/(n-4*k)!);

a(n) = (3 * a(n-1) + n * a(n-4) + 1)/4 for n > 3.

A357570 a(n) = Sum_{k=0..floor(n/5)} (n-4k)!/(n-5k)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 60, 93, 146, 225, 336, 509, 798, 1281, 2060, 3261, 5154, 8273, 13536, 22365, 36806, 60369, 99588, 166301, 280650, 474801, 802424, 1358973, 2317806, 3987185, 6893196, 11933949, 20690738, 36022161, 63107520, 111146141, 196322454, 347412753

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A072374, A122852, A357532, A357533.

a(n) = sum(k=0, n\5, (n-4*k)!/(n-5*k)!);

D-finite with recurrence a(n) = (4 * a(n-1) + n * a(n-5) + 1)/5 for n > 4.

A113288 Matrix inverse of triangle A113287.

Original entry on oeis.org

1, -2, 1, 3, 0, 1, -8, -4, -4, 1, 15, 10, 10, 0, 1, -36, -30, -36, -12, -6, 1, 77, 70, 91, 42, 21, 0, 1, -192, -184, -256, -152, -96, -24, -8, 1, 459, 450, 648, 432, 306, 108, 36, 0, 1, -1220, -1210, -1780, -1280, -1000, -460, -200, -40, -10, 1, 3201, 3190, 4741, 3542, 2926, 1540, 770, 220, 55, 0, 1

Offset: 0

Paul D. Hanna, Oct 23 2005

			Triangle begins:
.1;
.-2,1;
.3,0,1;
.-8,-4,-4,1;
.15,10,10,0,1;
.-36,-30,-36,-12,-6,1;
.77,70,91,42,21,0,1;
.-192,-184,-256,-152,-96,-24,-8,1;
.459,450,648,432,306,108,36,0,1;
.-1220,-1210,-1780,-1280,-1000,-460,-200,-40,-10,1;
.3201,3190,4741,3542,2926,1540,770,220,55,0,1; ...

Cf. A113287, A113289 (row sums), A113290 (-log), A072374.

{T(n,k)=local(x=X+O(X^(n+2)),y=Y+O(Y^(n+2)),M=matrix(n+1,n+1,r,c, polcoeff(polcoeff(1/(1-x*y)+r*x/((1-x*y)*(1+x+x*y)),r-1,X),c-1,Y))); if(n

T(n, 0) = (-1)^n*(n+1)*A072374(n-1) for n>=2, with T(1, 0)=-2, T(n, n)=1. T(n, 1) = (-1)^n*(n+1)*(A072374(n-1) - 1) for n>=2.

A113291 a(n) = A113290(n,1)/(n+1) for n>=0, where A113290 is the matrix log of triangle A113287.

Original entry on oeis.org

0, 0, 0, 1, -2, 4, -7, 13, -24, 48, -99, 221, -512, 1268, -3247, 8773, -24400, 70896, -211347, 653541, -2068472, 6755684, -22541135, 77305981, -270435640, 969413776, -3539893923, 13212871629, -50180362320, 194412817844, -765590169935, 3070433223317

Offset: 0

Paul D. Hanna, Oct 23 2005

sign

Cf. A113287, A113288, A113290, A072374.

a(n)=if(n<3,0,(-1)^(n-3)*sum(k=0,n-3,sum(j=0,k\2,(k-j)!/(k-2*j)!)))

G.f. satisfies: A(x) = x^3*((2+x)/(1+x) + (1+x)*A'(x))/(2+3*x+2*x^2). a(n+3) = (-1)^n*Sum_{k=0..n} Sum{j=0..[k/2]} (k-j)!/(k-2*j)! for n>=0. a(n+3) = -a(n+2) + (-1)^n*A072374(n) for n>=1.

cp's OEIS Frontend

A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A122852 Row sums of number triangle A122851.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2*k)!/(n-3*k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A113287 Triangle T, read by rows, where row n of T equals row n of matrix (n+1)-th power of triangle A112555.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A113290 Matrix logarithm of triangle A113287.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A357533 a(n) = Sum_{k=0..floor(n/4)} (n-3*k)!/(n-4*k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A357570 a(n) = Sum_{k=0..floor(n/5)} (n-4*k)!/(n-5*k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A113288 Matrix inverse of triangle A113287.

Views

Author

Keywords

Examples

Crossrefs

Programs

A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2k)!/(n-3k)!.

A357533 a(n) = Sum_{k=0..floor(n/4)} (n-3k)!/(n-4k)!.

A357570 a(n) = Sum_{k=0..floor(n/5)} (n-4k)!/(n-5k)!.