A007531 -id:A007531 - cp's OEIS frontend

A269467 T(n,k)=Number of length-n 0..k arrays with no repeated value equal to the previous repeated value.

2, 3, 4, 4, 9, 6, 5, 16, 24, 10, 6, 25, 60, 66, 14, 7, 36, 120, 228, 174, 22, 8, 49, 210, 580, 852, 462, 30, 9, 64, 336, 1230, 2780, 3180, 1206, 46, 10, 81, 504, 2310, 7170, 13300, 11796, 3150, 62, 11, 100, 720, 3976, 15834, 41730, 63420, 43644, 8166, 94, 12, 121

Offset: 1

R. H. Hardin, Feb 27 2016

Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
.10....66....228.....580.....1230......2310......3976.......6408.......9810
.14...174....852....2780.....7170.....15834.....31304......56952......97110
.22...462...3180...13300....41730....108402....246232.....505800.....960750
.30..1206..11796...63420...242370....741090...1934856....4488696....9499590
.46..3150..43644..301780..1405530...5060706..15190840...39808584...93880710
.62..8166.160980.1433180..8139570..34523202.119174216..352838520..927352710
.94.21150.592572.6795700.47082330.235304034.934305400.3125681352.9156504150
The conjectures regarding the recursions for column k are correct (see links) - Sela Fried, Oct 29 2024.

			Some solutions for n=6 k=4
..2. .0. .3. .1. .1. .1. .2. .0. .1. .0. .3. .0. .3. .2. .4. .1
..4. .3. .1. .1. .2. .0. .0. .2. .0. .2. .3. .1. .4. .0. .3. .4
..3. .2. .0. .0. .0. .0. .2. .2. .1. .0. .4. .2. .4. .0. .4. .3
..3. .0. .4. .2. .1. .1. .2. .1. .3. .4. .1. .4. .2. .4. .4. .1
..1. .3. .4. .4. .0. .4. .4. .3. .4. .1. .4. .1. .2. .2. .0. .0
..0. .1. .3. .3. .4. .3. .2. .4. .1. .3. .2. .2. .1. .4. .1. .2

R. H. Hardin, Table of n, a(n) for n = 1..9999
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024.

Column 1 is A027383.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
k=2: a(n) = 3*a(n-1) +2*a(n-2) -8*a(n-3)
k=3: a(n) = 5*a(n-1) -18*a(n-3)
k=4: a(n) = 7*a(n-1) -4*a(n-2) -32*a(n-3)
k=5: a(n) = 9*a(n-1) -10*a(n-2) -50*a(n-3)
k=6: a(n) = 11*a(n-1) -18*a(n-2) -72*a(n-3)
k=7: a(n) = 13*a(n-1) -28*a(n-2) -98*a(n-3)
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + 4*n^2 + n
n=5: a(n) = n^5 + 5*n^4 + 7*n^3 + 2*n^2 - n
n=6: a(n) = n^6 + 6*n^5 + 11*n^4 + 4*n^3 - n^2 + n
n=7: a(n) = n^7 + 7*n^6 + 16*n^5 + 8*n^4 - n^3 - n

A269537 T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by other than one.

Original entry on oeis.org

2, 3, 4, 4, 9, 6, 5, 16, 24, 10, 6, 25, 60, 64, 14, 7, 36, 120, 222, 164, 22, 8, 49, 210, 568, 804, 418, 30, 9, 64, 336, 1210, 2648, 2878, 1048, 46, 10, 81, 504, 2280, 6890, 12214, 10192, 2614, 62, 11, 100, 720, 3934, 15324, 38878, 55836, 35812, 6468, 94, 12, 121

Offset: 1

R. H. Hardin, Feb 29 2016

Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
.10....64....222.....568.....1210......2280......3934.......6352.......9738
.14...164....804....2648.....6890.....15324.....30464......55664......95238
.22...418...2878...12214....38878....102202....234358.....485038.....926854
.30..1048..10192...55836...217714....677200...1792788....4205812....8981446
.46..2614..35812..253418..1211476...4462414..13648124...36313762...86704348
.62..6468.125012.1143256..6705102..29265308.103462888..312366672..834223586
.94.15942.434110.5131592.36939610.191134204.781425950.2678039200.8002547722

			Some solutions for n=6 k=4
..2. .4. .4. .0. .3. .0. .0. .3. .1. .2. .2. .3. .2. .1. .4. .0
..4. .3. .1. .3. .2. .2. .3. .0. .2. .0. .1. .0. .1. .3. .0. .2
..3. .2. .0. .1. .1. .4. .1. .0. .3. .3. .3. .1. .3. .0. .4. .2
..2. .1. .1. .2. .1. .0. .3. .2. .3. .4. .1. .0. .4. .2. .1. .4
..0. .3. .1. .2. .0. .1. .1. .4. .2. .1. .1. .1. .1. .4. .1. .0
..2. .1. .2. .4. .2. .1. .2. .0. .0. .2. .2. .2. .3. .2. .4. .4

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A027383.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
k=2: a(n) = 4*a(n-1) -a(n-2) -10*a(n-3) +6*a(n-4) +4*a(n-5)
k=3: a(n) = 7*a(n-1) -9*a(n-2) -23*a(n-3) +31*a(n-4) +33*a(n-5)
k=4: a(n) = 14*a(n-1) -65*a(n-2) +80*a(n-3) +163*a(n-4) -280*a(n-5) -208*a(n-6)
k=5: [order 7]
k=6: [order 9]
k=7: [order 9]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + 3*n^2 + 2*n
n=5: a(n) = n^5 + 5*n^4 + 4*n^3 + 6*n^2 - 2*n
n=6: a(n) = n^6 + 6*n^5 + 5*n^4 + 12*n^3 - 6*n^2 + 6*n - 2
n=7: a(n) = n^7 + 7*n^6 + 6*n^5 + 20*n^4 - 12*n^3 + 22*n^2 - 18*n + 4

A269606 T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

2, 3, 4, 4, 9, 6, 5, 16, 24, 8, 6, 25, 60, 62, 10, 7, 36, 120, 222, 154, 12, 8, 49, 210, 572, 804, 376, 14, 9, 64, 336, 1220, 2692, 2878, 902, 16, 10, 81, 504, 2298, 7030, 12570, 10192, 2142, 18, 11, 100, 720, 3962, 15630, 40288, 58280, 35812, 5040, 20, 12, 121

Offset: 1

R. H. Hardin, Mar 01 2016

Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
..8....62....222.....572.....1220......2298......3962.......6392.......9792
.10...154....804....2692.....7030.....15630.....31024......56584......96642
.12...376...2878...12570....40288....105892....242226.....499798.....952180
.14...902..10192...58280...229754....714874...1886252....4405772....9366790
.16..2142..35812..268704..1304934...4811578..14654952...38768412...92013754
.18..5040.125012.1233046..7385898..32300252.113629480..340600002..902743646
.20.11786.434110.5636046.41679780.216337084.879470154.2988094770.8846649136

			Some solutions for n=6 k=4
..1. .0. .3. .0. .3. .2. .1. .1. .0. .3. .2. .3. .2. .1. .1. .2
..3. .2. .2. .2. .0. .4. .2. .2. .3. .4. .0. .0. .0. .4. .1. .2
..0. .0. .3. .4. .3. .2. .3. .3. .1. .4. .4. .3. .3. .3. .2. .3
..4. .1. .0. .3. .1. .0. .3. .1. .0. .3. .1. .1. .3. .0. .1. .1
..3. .3. .2. .4. .3. .4. .0. .0. .1. .4. .2. .3. .2. .3. .2. .0
..1. .3. .1. .3. .0. .4. .1. .3. .1. .0. .3. .3. .0. .3. .0. .0

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A004275(n+1).
Column 3 is A269532.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-2)
k=2: a(n) = 5*a(n-1) -5*a(n-2) -8*a(n-3) +12*a(n-4)
k=3: a(n) = 7*a(n-1) -9*a(n-2) -23*a(n-3) +31*a(n-4) +33*a(n-5)
k=4: [order 7]
k=5: [order 7]
k=6: [order 9]
k=7: [order 9]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + 4*n^2 - n
n=5: a(n) = n^5 + 5*n^4 + 7*n^3 - 4*n^2 + n
n=6: a(n) = n^6 + 6*n^5 + 11*n^4 - 8*n^3 + n^2 + 3*n - 2
n=7: a(n) = n^7 + 7*n^6 + 16*n^5 - 12*n^4 - 5*n^3 + 18*n^2 - 15*n + 4

A269640 T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by other than plus two or minus 1.

Original entry on oeis.org

2, 3, 4, 4, 9, 6, 5, 16, 24, 9, 6, 25, 60, 63, 12, 7, 36, 120, 221, 159, 16, 8, 49, 210, 567, 796, 396, 20, 9, 64, 336, 1209, 2637, 2828, 969, 25, 10, 81, 504, 2279, 6876, 12125, 9928, 2349, 30, 11, 100, 720, 3933, 15307, 38738, 55225, 34537, 5640, 36, 12, 121, 990

Offset: 1

R. H. Hardin, Mar 02 2016

Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
..9....63....221.....567.....1209......2279......3933.......6351.......9737
.12...159....796....2637.....6876.....15307.....30444......55641......95212
.16...396...2828...12125....38738....101999....234080.....484673.....926390
.20...969...9928...55225...216528....675151...1789528....4200933....8974480
.25..2349..34537..249600..1202353...4443665..13613507...36254755...86609789
.30..5640.119236.1120868..6639294..29104549.103118640..311698647..833022466
.36.13455.409098.5006144.36486190.189818232.778158768.2670823421.7987993868

			Some solutions for n=6 k=4
..4. .2. .3. .4. .3. .0. .2. .4. .0. .3. .1. .0. .4. .1. .0. .3
..0. .3. .1. .2. .1. .0. .1. .3. .0. .4. .2. .2. .1. .3. .3. .2
..3. .1. .2. .4. .4. .3. .4. .0. .1. .3. .0. .1. .0. .2. .0. .4
..2. .4. .0. .3. .2. .1. .1. .2. .2. .0. .1. .0. .3. .0. .3. .0
..1. .1. .2. .3. .2. .2. .3. .0. .4. .3. .0. .3. .2. .4. .1. .1
..0. .4. .3. .0. .4. .1. .3. .2. .2. .0. .2. .0. .4. .1. .2. .1

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A002620(n+2).
Column 2 is A268938.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)
k=2: a(n) = 3*a(n-1) +a(n-2) -6*a(n-3)
k=3: a(n) = 9*a(n-1) -21*a(n-2) -19*a(n-3) +93*a(n-4) +27*a(n-5) -133*a(n-6) -87*a(n-7)
k=4: [order 7]
k=5: [order 13]
k=6: [order 14]
k=7: [order 16]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + 3*n^2 + 2*n - 1
n=5: a(n) = n^5 + 5*n^4 + 4*n^3 + 6*n^2 - 5*n + 1
n=6: a(n) = n^6 + 6*n^5 + 5*n^4 + 12*n^3 - 12*n^2 + 9*n - 7 for n>2
n=7: a(n) = n^7 + 7*n^6 + 6*n^5 + 20*n^4 - 22*n^3 + 28*n^2 - 37*n + 13 for n>2

A269678 T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by other than plus or minus one modulo k+1.

Original entry on oeis.org

2, 3, 4, 4, 9, 6, 5, 16, 24, 10, 6, 25, 60, 66, 14, 7, 36, 120, 224, 174, 22, 8, 49, 210, 570, 820, 462, 30, 9, 64, 336, 1212, 2670, 2976, 1206, 46, 10, 81, 504, 2282, 6918, 12390, 10700, 3150, 62, 11, 100, 720, 3936, 15358, 39156, 57030, 38224, 8166, 94, 12, 121

Offset: 1

R. H. Hardin, Mar 03 2016

Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
.10....66....224.....570.....1212......2282......3936.......6354.......9740
.14...174....820....2670.....6918.....15358.....30504......55710......95290
.22...462...2976...12390....39156....102606....234912.....485766.....927780
.30..1206..10700...57030...220050....681254...1799256....4215510....8995310
.46..3150..38224..260790..1229292...4499278..13716480...36430614...86891980
.62..8166.135780.1185990..6832518..29579382.104139336..313684470..836599530
.94.21150.480176.5368470.37810116.193688894.787814400.2692218006.8031245540

			Some solutions for n=6 k=4
..0. .1. .4. .0. .2. .4. .0. .4. .4. .4. .4. .4. .3. .0. .4. .4
..2. .3. .2. .3. .3. .1. .0. .2. .4. .2. .0. .3. .3. .3. .0. .2
..0. .2. .4. .2. .4. .3. .4. .1. .0. .3. .3. .2. .0. .0. .3. .3
..3. .3. .1. .1. .1. .4. .1. .2. .2. .3. .2. .1. .4. .4. .3. .4
..0. .3. .3. .0. .4. .2. .0. .3. .0. .2. .1. .0. .3. .4. .4. .1
..1. .2. .4. .2. .4. .2. .4. .1. .4. .0. .2. .1. .4. .3. .2. .3

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A027383.
Column 2 is A269461.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
k=2: a(n) = 3*a(n-1) +2*a(n-2) -8*a(n-3)
k=3: a(n) = 5*a(n-1) -a(n-2) -15*a(n-3)
k=4: a(n) = 7*a(n-1) -6*a(n-2) -24*a(n-3)
k=5: a(n) = 9*a(n-1) -13*a(n-2) -35*a(n-3)
k=6: a(n) = 11*a(n-1) -22*a(n-2) -48*a(n-3)
k=7: a(n) = 13*a(n-1) -33*a(n-2) -63*a(n-3)
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + 3*n^2 + 2*n + 2 for n>1
n=5: a(n) = n^5 + 5*n^4 + 4*n^3 + 6*n^2 + 4*n - 2 for n>1
n=6: a(n) = n^6 + 6*n^5 + 5*n^4 + 12*n^3 + 6*n^2 + 6 for n>1
n=7: a(n) = n^7 + 7*n^6 + 6*n^5 + 20*n^4 + 8*n^3 + 10*n^2 + 12*n - 10 for n>1

A135503 a(n) = n*(n^2 - 1)/2.

Original entry on oeis.org

0, 0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

Offset: 0

Cino Hilliard, Feb 09 2008

Previous name was: Integer values of sqrt(b) solving sqrt(d) + sqrt(b) = sqrt(c) with d^2 + b = c.
Squaring the first equation and setting the result equal to the second, we need d + b + 2*sqrt(d*b) = d^2+b -> d + 2*sqrt(d*b) = d^2 -> d^2 - d = 2*sqrt(d*b)
-> d^2*(d-1)^2 = 4*d*b -> b = d*(d-1)^2/4 -> sqrt(b) = (d-1)*sqrt(d)/2. Setting d = (n+1)^2 yields sqrt(b) = A027480(n).
This is the case k = 2 for FLTR, Fermat's Last Theorem with rational exponents 1/k: Consider x + y = x + y. Then (x^k)^(1/k) + (y^k)^(1/k) = ((x+y)^k)^(1/k).
For k > 2, there are infinitely many solutions to d^(1/k) + b^(1/k) = c^(1/k). E.g., 8^(1/3) + 27^(1/3) = 125^(1/3) at k = 3. However, in conjunction with d^2 + b = c, I could not find any nontrivial solutions.
A shifted version of A027480. - R. J. Mathar, Apr 07 2009
For n > 2, a(n) is the maximum value of the magic constant in a perimeter-magic n-gon of order n (see A342758). - Stefano Spezia, Mar 21 2021
a(n) is equal to the total number of P_3 edge-disjoint subgraphs of the complete graph on n vertices. - Samuel J. Bevins, May 09 2023

			For d = 9, b = 144, c = 225, 9^(1/2) + 144^(1/2) = 225^(1/2) and 9^2 + 144 = 225. So b^(1/2) = 12 is the 4th entry in the sequence.

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. D. Bennet, A. M. W. Glass and G. J. Székely, Fermat's Last Theorem for Rational Exponents, Am. Math. Monthly 111 (2004), 322-329. - _R. J. Mathar_, Apr 21 2009
Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20.

Cf. A000217, A000578, A002411, A004188, A006003, A007531, A007588, A027480, A342758.

Array[# (#^2 - 1)/2 &, 42, 0] (* Michael De Vlieger, Feb 20 2018 *)

flt2(n,p) = { local(a,b); for(a=0,n, b = (a^3-a)/2; print1(b", ") ) }

a(n) = 3*A000292(n-1).
From R. J. Mathar Feb 20 2008: (Start)
O.g.f.: 3*x^2/(-1+x)^4.
a(n) = n*(n^2 - 1)/2 = A007531(n+1)/2. (End)
G.f.: 3*x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
a(n) = A006003(n+1) - A000326(n+1). - J. M. Bergot, Dec 04 2014
E.g.f.: (1/2)* x^2 *(3 + x)*exp(x). - G. C. Greubel, Oct 15 2016
From Miquel Cerda, Dec 25 2016: (Start)
a(n) = A000578(n) - A006003(n).
a(n) = A004188(n) - A000578(n).
a(n) = A007588(n) - A004188(n). (End)
a(n) = A002411(n) - A000217(n). - Justin Gaetano, Feb 20 2018
From Amiram Eldar, Jan 09 2021: (Start)
Sum_{n>=2} 1/a(n) = 1/2.
Sum_{n>=2} (-1)^n/a(n) = 4*log(2) - 5/2. (End)

Edited by R. J. Mathar, Apr 21 2009

New name using R. J. Mathar's formula, Joerg Arndt, Dec 05 2014

A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 4, 0, 2, 0, 5, 0, 6, 2, 0, 6, 0, 12, 6, 6, 0, 7, 0, 20, 12, 24, 4, 0, 8, 0, 30, 20, 60, 18, 14, 0, 9, 0, 42, 30, 120, 48, 78, 12, 0, 10, 0, 56, 42, 210, 100, 252, 72, 28, 0, 11, 0, 72, 56, 336, 180, 620, 240, 234, 24, 0, 12, 0, 90, 72, 504, 294, 1290, 600, 1008, 216, 62

Offset: 1

Andrew Howroyd, Apr 03 2017

			Table starts:
1  2   3    4    5    6     7     8     9    10 ...
0  0   0    0    0    0     0     0     0     0 ...
0  2   6   12   20   30    42    56    72    90 ...
0  2   6   12   20   30    42    56    72    90 ...
0  6  24   60  120  210   336   504   720   990 ...
0  4  18   48  100  180   294   448   648   900 ...
0 14  78  252  620 1290  2394  4088  6552  9990 ...
0 12  72  240  600 1260  2352  4032  6480  9900 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 24 216  960 3000 7560 16464 32256 58320 99000 ...
...
Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2-6 are A056458, A056459, A056460, A056461, A056462.
Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.
Cf. A284826, A284841.

T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017 *)

a(n,k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print();)

T(n,k) = Sum_{d | n} mu(n/d) * k^(ceiling(d/2)).

A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

Original entry on oeis.org

1, -2, 6, 3, -24, 30, -4, 60, -180, 140, 5, -120, 630, -1120, 630, -6, 210, -1680, 5040, -6300, 2772, 7, -336, 3780, -16800, 34650, -33264, 12012, -8, 504, -7560, 46200, -138600, 216216, -168168, 51480, 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790

Offset: 0

N. J. A. Sloane, Jan 17 2020

Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.

Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - Michael Somos, Mar 20 2020 [Corrected by Petros Hadjicostas, Jul 09 2020]

			Triangle begins:
   1;
  -2,    6;
   3,  -24,    30;
  -4,   60,  -180,     140;
   5, -120,   630,   -1120,     630;
  -6,  210, -1680,    5040,   -6300,     2772;
   7, -336,  3780,  -16800,   34650,   -33264,   12012;
  -8,  504, -7560,   46200, -138600,   216216, -168168,   51480;
   9, -720, 13860, -110880,  450450, -1009008, 1261260, -823680, 218790;
  ...

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Buhl, Book review: J. Ser - Les calculs formels des séries de factorielles, L'Enseignement Mathématique, 32 (1933), p. 275.
L. A. MacColl, Review: J. Ser, Les calculs formels des séries de factorielles, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.
L. M. Milne-Thomson, Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars), The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages.)

Columns 1 is A331433 or equally A007531, column 2 is A331434 or equally A054559; the last three diagonals are A002738, A002736, A002457.

Cf. A000290 (row sums), A002457,, A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.

[(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1,n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022

gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
seq(seq(T(n,k), k=0..n),n=0..7); # Peter Luschny, Jan 18 2020
S:=(n,k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
rho:=n->[seq(S(n,k),k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od: # N. J. A. Sloane, Jan 18 2020

Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k,k]*Binomial[n+k,n-k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)

flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k,k)*binomial(n+k+1,n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022

T(n, 0) = (-1)^n*A000027(n+1).
T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).
T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).
T(n, n-2) = A002738(n-2).
T(n, n-1) = (-1)*A002736(n).
T(n, n) = A002457(n).
T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).
Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).
Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).
2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).
(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).
T(n, k) = ((2*k+1)!/(k!)^2)*[x^(n-k)] (1+x)^(-2*(k+1)). - Georg Fischer and Peter Luschny, Jan 18 2020
T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - N. J. A. Sloane, Jan 18 2020
From Petros Hadjicostas, Jul 09 2020: (Start)
Michael Somos's formulas above can be restated as
Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.
These are special cases of the following formula that is alluded to (in some way) in Ser's book:
1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).
Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)
T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022

Several typos in the data corrected by Georg Fischer and Peter Luschny, Jan 18 2020

Definition changed by N. J. A. Sloane, Jan 18 2020

A053625 Product of 6 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 5040, 20160, 60480, 151200, 332640, 665280, 1235520, 2162160, 3603600, 5765760, 8910720, 13366080, 19535040, 27907200, 39070080, 53721360, 72681840, 96909120, 127512000, 165765600, 213127200, 271252800, 342014400, 427518000, 530122320

Offset: 0

Henry Bottomley, Mar 20 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A007531, A045619, A052762, A052787.

Cf. A000579, A173333.

F:=Factorial;; Concatenation([0,0,0,0,0,0], List([6..30], n-> F(n)/F(n-5) )); # G. C. Greubel, Aug 27 2019

I:=[0,0,0,0,0,0,720]; [n le 7 select I[n] else 7*Self(n-1) -21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6) +Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 28 2012

seq(combinat[numbperm](n, 6), n=0..31); # Zerinvary Lajos, Apr 26 2007

CoefficientList[Series[720*x^6/(1-x)^7,{x,0,30}],x] (* Vincenzo Librandi, Apr 28 2012 *)
Times@@@Partition[Range[-5,30],6,1] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,0,0,720},30] (* Harvey P. Dale, Nov 13 2015 *)
Pochhammer[Range[30]-6, 6] (* G. C. Greubel, Aug 27 2019 *)

a(n)=factorback([n-5..n]) \\ Charles R Greathouse IV, Oct 07 2015

[rising_factorial(n-5,6) for n in (0..30)] # G. C. Greubel, Aug 27 2019

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5) = n!/(n-6)! = A052787(n)*(n-6) = a(n-1)*n/(n-6).
E.g.f.: x^6*exp(x).
a(n) = 720 * A000579(n). - Zerinvary Lajos, Apr 26 2007
For n > 5: a(n) = A173333(n, n-6). - Reinhard Zumkeller, Feb 19 2010
G.f.: 720*x^6/(1-x)^7. - Colin Barker, Mar 27 2012
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Vincenzo Librandi, Apr 28 2012
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=6} 1/a(n) = 1/600.
Sum_{n>=6} (-1)^n/a(n) = 4*log(2)/15 - 661/3600. (End)

A128964 a(n) = (n^3-n)*6^n.

Original entry on oeis.org

0, 216, 5184, 77760, 933120, 9797760, 94058496, 846526464, 7255941120, 59861514240, 478892113920, 3735358488576, 28524555730944, 213934167982080, 1579821548175360, 11510128422420480, 82872924641427456, 590469588070170624, 4168020621671792640, 29176144351702548480

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (24,-216,864,-1296).

Cf. A000400, A007531, A036289, A081144, A128796.

Cf. A128960, A128961, A128962, A128963, A128965, A128967, A128969.

[(n^3-n)*6^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

I:=[0, 216, 5184, 77760]; [n le 4 select I[n] else 24*Self(n-1) -216*Self(n-2) +864*Self(n-3) -1296*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

CoefficientList[Series[216 x/(1 - 6 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 216*x^2/(1-6*x)^4.
a(n) = 216*A081144(n+1). (End)
a(n) = 24*a(n-1) - 216*a(n-2) + 864*a(n-3) - 1296*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=2} 1/a(n) = 25*log(6/5)/12 - 3/8.
Sum_{n>=2} (-1)^n/a(n) = 49*log(7/6)/12 - 5/8. (End)
a(n) = A007531(n+1)*A000400(n). - Amiram Eldar, Oct 02 2022

Corrected offset. - Mohammad K. Azarian, Nov 20 2008

cp's OEIS Frontend

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A269606 T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by one or less.

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A269640 T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by other than plus two or minus 1.

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A269678 T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by other than plus or minus one modulo k+1.

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A135503 a(n) = n*(n^2 - 1)/2.

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A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

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A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.

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A331431 Triangle read by rows: T(n,k) = (-1)^(n+k)(n+k+1)binomial(n,k)*binomial(n+k,k) for n >= k >= 0.