cp's OEIS Frontend

Empirical for column k:

k=1: a(n) = 2*n

k=2: a(n) = 4*n^2

k=3: a(n) = 12*n^2 - 6*n

k=4: a(n) = 6*n^3 + (27/2)*n^2 - (21/2)*n

k=5: a(n) = (13/6)*n^4 + 13*n^3 + (52/3)*n^2 - (39/2)*n

k=6: a(n) = (33/4)*n^4 + (45/2)*n^3 + (75/4)*n^2 - (63/2)*n

k=7: a(n) = (55/24)*n^5 + (75/4)*n^4 + (275/8)*n^3 + (75/4)*n^2 - (295/6)*n

Empirical for column k:

k=1: a(n) = 2*n

k=2: a(n) = 4*n^2

k=3: a(n) = 2*n^3 + 6*n^2 - 2*n

k=4: a(n) = (3/4)*n^4 + (15/2)*n^3 + (15/4)*n^2 - 3*n

k=5: a(n) = (7/30)*n^5 + 7*n^4 + (21/2)*n^3 - (56/15)*n

k=6: a(n) = (7/120)*n^6 + (147/40)*n^5 + (49/3)*n^4 + (91/8)*n^3 - (707/120)*n^2 - (91/20)*n

k=7: a(n) = (31/2520)*n^7 + (62/45)*n^6 + (4867/360)*n^5 + (2015/72)*n^4 + (1271/180)*n^3 - (4991/360)*n^2 - (713/140)*n

Empirical for column k:

k=1: a(n) = 2*n

k=2: a(n) = 4*n^2

k=3: a(n) = (4/3)*n^3 + 8*n^2 - (10/3)*n

k=4: a(n) = (5/12)*n^4 + (13/2)*n^3 + (115/12)*n^2 - (17/2)*n + 1

k=5: a(n) = (7/60)*n^5 + (8/3)*n^4 + (185/12)*n^3 + (19/3)*n^2 - (218/15)*n + 3

k=6: a(n) = (7/360)*n^6 + (77/120)*n^5 + (635/72)*n^4 + (623/24)*n^3 - (511/180)*n^2 - (103/5)*n + 6

k=7: a(n) = (1/280)*n^7 + (7/45)*n^6 + (47/15)*n^5 + (206/9)*n^4 + (4111/120)*n^3 - (1037/45)*n^2 - (4493/210)*n + 9

Empirical for column k:

k=1: a(n) = 2*n

k=2: a(n) = 4*n^2

k=3: a(n) = 12*n^2 - 6*n

k=4: a(n) = 9*n^3 + 9*n - 9

k=5: a(n) = (13/4)*n^4 + (13/2)*n^3 + (117/4)*n^2 - 26*n

k=6: a(n) = (19/4)*n^4 + (95/2)*n^3 - (57/4)*n^2 - 19*n

k=7: a(n) = 35*n^4 + 42*n^3 + 7*n^2 - 84*n + 28

Empirical for rows:

n=1: a(k)=a(k-1)+a(k-3) for k>4

n=2: a(k)=a(k-1)+a(k-2)+3*a(k-3)+a(k-4)-a(k-5)-a(k-6) for k>7

n=3: a(k)=a(k-1)+9*a(k-3)+2*a(k-4)+2*a(k-5)-12*a(k-6)-8*a(k-7)+8*a(k-9) for k>11

n=4: a(k)=a(k-1)+13*a(k-3)+3*a(k-4)+3*a(k-5)-27*a(k-6)-18*a(k-7)+27*a(k-9) for k>11

n=5: a(k)=a(k-1)+17*a(k-3)+4*a(k-4)+4*a(k-5)-48*a(k-6)-32*a(k-7)+64*a(k-9) for k>11

n=6: a(k)=a(k-1)+21*a(k-3)+5*a(k-4)+5*a(k-5)-75*a(k-6)-50*a(k-7)+125*a(k-9) for k>11

n=7: a(k)=a(k-1)+25*a(k-3)+6*a(k-4)+6*a(k-5)-108*a(k-6)-72*a(k-7)+216*a(k-9) for k>11

n=8: a(k)=a(k-1)+29*a(k-3)+7*a(k-4)+7*a(k-5)-147*a(k-6)-98*a(k-7)+343*a(k-9) for k>11

n=9: a(k)=a(k-1)+33*a(k-3)+8*a(k-4)+8*a(k-5)-192*a(k-6)-128*a(k-7)+512*a(k-9) for k>11

n=10: a(k)=a(k-1)+37*a(k-3)+9*a(k-4)+9*a(k-5)-243*a(k-6)-162*a(k-7)+729*a(k-9) for k>11

n=11: a(k)=a(k-1)+41*a(k-3)+10*a(k-4)+10*a(k-5)-300*a(k-6)-200*a(k-7)+1000*a(k-9) for k>11

n=12: a(k)=a(k-1)+45*a(k-3)+11*a(k-4)+11*a(k-5)-363*a(k-6)-242*a(k-7)+1331*a(k-9) for k>11

n=13: a(k)=a(k-1)+49*a(k-3)+12*a(k-4)+12*a(k-5)-432*a(k-6)-288*a(k-7)+1728*a(k-9) for k>11

n=14: a(k)=a(k-1)+53*a(k-3)+13*a(k-4)+13*a(k-5)-507*a(k-6)-338*a(k-7)+2197*a(k-9) for k>11

n=15: a(k)=a(k-1)+57*a(k-3)+14*a(k-4)+14*a(k-5)-588*a(k-6)-392*a(k-7)+2744*a(k-9) for k>11

apparently a(k)=a(k-1)+(4*n-3)*a(k-3)+(n-1)*a(k-4)+(n-1)*a(k-5)-3*(n-1)^2*a(k-6)-2*(n-1)^2*a(k-7)+(n-1)^3*a(k-9) for n>2 and k>11

a(n) = 2*n*(n^2 + 3*n - 1) = 2*n*A014209(n). More generally, number of m X n (0, 1) matrices such that each row and each column is increasing or decreasing is 2*n*(2*binomial(n+m-1, n)-m) = 2*m*(2*binomial(m+n-1, m)-n).

G.f.: 6*x*(1 + 2*x - x^2)/(1-x)^4. - Vincenzo Librandi, Jun 24 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 24 2012

a(n) = 20*A000290(n) = 10*A001105(n) = 5*A016742(n) = 4*A033429(n) = 2*A033583(n).

a(0)=0, a(1)=20, a(2)=80; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 18 2013

a(n) = A010014(n) - A005899(n) for n > 0. - R. J. Cano, Sep 29 2015

From Elmo R. Oliveira, Nov 30 2024: (Start)

G.f.: 20*x*(1 + x)/(1-x)^3.

E.g.f.: 20*x*(1 + x)*exp(x).

a(n) = n*A008602(n) = A195148(2*n). (End)

Empirical for row n:

n=1: a(k) = 2*k

n=2: a(k) = 4*k^2

n=3: a(k) = 2*k^3 + 6*k^2 - 2*k

n=4: a(k) = (5/6)*k^4 + (35/3)*k^3 - (5/6)*k^2 - (5/3)*k

n=5: a(k) = (4/15)*k^5 + (32/3)*k^4 + (44/3)*k^3 - (32/3)*k^2 + (16/15)*k

n=6: a(k) = (13/180)*k^6 + (143/20)*k^5 + (1235/36)*k^4 - (39/4)*k^3 - (377/45)*k^2 + (13/5)*k

n=7: a(k) = (1/60)*k^7 + (77/20)*k^6 + (2527/60)*k^5 + (119/4)*k^4 - (644/15)*k^3 + (42/5)*k^2 + (4/5)*k

Empirical for column k:

k=1: a(n) = 2*n;

k=2: a(n) = 4*n^2;

k=3: a(n) = 12*n^2 - 6*n;

k=4: a(n) = 10*n^3 + 10*n^2 - 10*n;

k=5: a(n) = 48*n^3 - 32*n^2;

k=6: a(n) = 26*n^4 + 78*n^3 - 104*n^2 + 26*n;

k=7: a(n) = 168*n^4 - 84*n^3 - 84*n^2 + 42*n;

k=8: a(n) = 68*n^5 + 408*n^4 - 612*n^3 + 204*n^2;

k=9: a(n) = 550*n^5 - 990*n^3 + 660*n^2 - 110*n;

k=10: a(n) = 178*n^6 + 1780*n^5 - 2670*n^4 + 534*n^3 + 534*n^2 - 178*n;

k=11: a(n) = 1728*n^6 + 1440*n^5 - 6912*n^4 + 5184*n^3 - 1152*n^2;

k=12: a(n) = 466*n^7 + 6990*n^6 - 9320*n^5 - 2796*n^4 + 8388*n^3 - 3728*n^2 + 466*n;

k=13: a(n) = 5278*n^7 + 10556*n^6 - 36946*n^5 + 27144*n^4 - 3016*n^3 - 3016*n^2 + 754*n;

k=14: a(n) = 1220*n^8 + 25620*n^7 - 25620*n^6 - 42700*n^5 + 73200*n^4 - 36600*n^3 + 6100*n^2;

k=15: a(n) = 15792*n^8 + 55272*n^7 - 165816*n^6 + 98700*n^5 + 39480*n^4 - 59220*n^3 + 19740*n^2 - 1974*n.

Empirical for row n:

n=1: a(k)=a(k-1)+a(k-2);

n=2: a(k)=2*a(k-1)+2*a(k-2)-a(k-3);

n=3: a(k)=a(k-1)+7*a(k-2)+2*a(k-3)-4*a(k-4);

n=4: a(k)=a(k-1)+10*a(k-2)+3*a(k-3)-9*a(k-4);

n=5: a(k)=a(k-1)+13*a(k-2)+4*a(k-3)-16*a(k-4);

n=6: a(k)=a(k-1)+16*a(k-2)+5*a(k-3)-25*a(k-4);

n=7: a(k)=a(k-1)+19*a(k-2)+6*a(k-3)-36*a(k-4);

apparently for row n>2: a(k)=a(k-1)+(3*n-2)*a(k-2)+(n-1)*a(k-3)+(n-1)^2*a(k-4).

a(n) = a(n-1) + 8*n + 2. - Vincenzo Librandi, Aug 08 2010

From Michael De Vlieger, Jun 15 2017: (Start)

G.f.: (1 + 8*x - x^2)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

a(n) = A016742(n) + A008588(n) + 1. - Felix Fröhlich, Jun 16 2017

Sum_{k=1..n} a(k-1)/(2*k)! = 1 - 1/(2*n)!. - Robert Israel, Jul 19 2017

E.g.f.: (1 + 10*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022

a(n) - a(n-1) = 4*(n+1)^2 = A016742(n+1).

a(n) - 2*a(n-1) + a(n-2) = -4 + 8*n = A017113(n+1).

a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 8 = A010731(n).

a(n) - 4*a(n-1) + 6*a(n-2) - 4*a(n-3) + a(n-4) = 0.

Binomial transform of quasi-finite sequence 1,4,12,8,0,(0 continued).

G.f.: (1+x+7*x^2-x^3)/(1-x)^4. - R. J. Mathar, Feb 15 2010

From Natan Arie Consigli, Jul 03 2016: (Start)

a(n) = A018227(2*n) + 3.

a(n) = A002492(n) + 1. (End)

E.g.f.: (1/3)*(3 + 12*x + 18*x^2 + 4*x^3)*exp(x). - G. C. Greubel, Jul 27 2024