A000125 -id:A000125 - cp's OEIS frontend

A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 15, 15, 11, 5, 6, 16, 26, 30, 26, 16, 6, 7, 22, 42, 56, 56, 42, 22, 7, 8, 29, 64, 98, 112, 98, 64, 29, 8, 9, 37, 93, 162, 210, 210, 162, 93, 37, 9, 10, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10

Offset: 0

Asher Auel

The number of spotlight tilings of an m X n rectangle missing the southeast corner. E.g., there are 2 spotlight tilings of a 2 X 2 square missing its southeast corner. - Bridget Tenner, Nov 10 2007
T(n,k) = A134636(n,k) - A051597(n,k). - Reinhard Zumkeller, Nov 23 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

			From _Roger L. Bagula_, Feb 17 2009: (Start)
Triangle begins:
   0;
   1,  1;
   2,  2,   2;
   3,  4,   4,   3;
   4,  7,   8,   7,    4;
   5, 11,  15,  15,   11,    5;
   6, 16,  26,  30,   26,   16,   6;
   7, 22,  42,  56,   56,   42,   22,    7;
   8, 29,  64,  98,  112,   98,   64,   29,   8;
   9, 37,  93, 162,  210,  210,  162,   93,   37,   9;
  10, 46, 130, 255,  372,  420,  372,  255,  130,  46,  10;
  11, 56, 176, 385,  627,  792,  792,  627,  385, 176,  56, 11;
  12, 67, 232, 561, 1012, 1419, 1584, 1419, 1012, 561, 232, 67, 12. ... (End)

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
B. E. Tenner, Spotlight tiling, Ann. Combinat. 14 (4) (2010) 553-568.
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A000918(n+1).
Cf. A007318, A224791, A228196, A228576.
Columns from 2 to 9, respectively: A000124; A000125, A055795, A027660, A055796, A055797, A055798, A055799 (except 1 for the last seven). [Bruno Berselli, Aug 02 2013]
Cf. A001477, A162551 (central terms).

Flat(List([0..12], n-> List([0..n], k->  Binomial(n, k+1) + Binomial(n, n-k+1) ))); # G. C. Greubel, Nov 12 2019

a051601 n k = a051601_tabl !! n !! k
a051601_row n = a051601_tabl !! n
a051601_tabl = iterate
               (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [0]
-- Reinhard Zumkeller, Nov 23 2012

/* As triangle: */ [[Binomial(n,m+1)+Binomial(n,n-m+1): m in [0..n]]: n in [0..12]]; // Bruno Berselli, Aug 02 2013

seq(seq(binomial(n,k+1) + binomial(n, n-k+1), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019

T[n_, k_]:= T[n, k] = Binomial[n, k+1] + Binomial[n, n-k+1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 17 2009; modified by G. C. Greubel, Nov 12 2019 *)

T(n,k) = binomial(n, k+1) + binomial(n, n-k+1);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 12 2019

[[binomial(n, k+1) + binomial(n, n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

T(m,n) = binomial(m+n,m) - 2*binomial(m+n-2,m-1), up to offset and transformation of array to triangular indices. - Bridget Tenner, Nov 10 2007
T(n,k) = binomial(n, k+1) + binomial(n, n-k+1). - Roger L. Bagula, Feb 17 2009
T(0,n) = T(n,0) = n, T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n.

A086514 Difference between the arithmetic mean of the neighbors of the terms and the term itself follows the pattern 0,1,2,3,4,5,...

Original entry on oeis.org

1, 2, 3, 6, 13, 26, 47, 78, 121, 178, 251, 342, 453, 586, 743, 926, 1137, 1378, 1651, 1958, 2301, 2682, 3103, 3566, 4073, 4626, 5227, 5878, 6581, 7338, 8151, 9022, 9953, 10946, 12003, 13126, 14317, 15578, 16911, 18318, 19801, 21362, 23003, 24726

Offset: 1

Amarnath Murthy, Jul 29 2003

{a(k): 1 <= k <= 4} = divisors of 6. - Reinhard Zumkeller, Jun 17 2009

			2 = (1+3)/2 -0. 3 = (2+6)/2 - 1, 6 = (3+13)/2 - 2, etc.

B. Berselli, Table of n, a(n) for n = 1..10000 - _Bruno Berselli_, May 31 2010
R. Zumkeller, Enumerations of Divisors - _Reinhard Zumkeller_, Jun 17 2009
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408, A000124, A016813, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261, A177342, A014106, A000290.

a(n) = n*(n^2-6*n+14)/3-2 \\ Charles R Greathouse IV, Jun 11 2015

a(n)+ n-2 = {a(n-1) +a(n+1)}/2
a(n) = (n^3-6*n^2+14*n-6)/3.
Contribution from Bruno Berselli, May 31 2010: (Start)
G.f.: (1-2*x+x^2+2*x^3)/(1-x)^4.
a(n)-4*a(n-1)+6*a(n-2)-4*a(n-3)+a(n-4) = 0 with n>4. For n=9, 121-4*78+6*47-4*26+13 = 0.
a(n) = ( A177342(n)-A000290(n-1)-3*A014106(n-2) )/4 with n>1. For n=11, a(11) = (1671-100-3*189)/4 = 251. (End)

More terms from David Wasserman, Mar 10 2005

A006000 a(n) = (n+1)(n^2+n+2)/2; g.f.: (1 + 2x^2) / (1 - x)^4.

Original entry on oeis.org

1, 4, 12, 28, 55, 96, 154, 232, 333, 460, 616, 804, 1027, 1288, 1590, 1936, 2329, 2772, 3268, 3820, 4431, 5104, 5842, 6648, 7525, 8476, 9504, 10612, 11803, 13080, 14446, 15904, 17457, 19108, 20860, 22716, 24679, 26752, 28938, 31240, 33661, 36204, 38872, 41668, 44595, 47656, 50854, 54192

Offset: 0

N. J. A. Sloane, Simon Plouffe

Enumerates certain paraffins.
a(n) is the (n+1)st (n+3)-gonal number. - Floor van Lamoen, Oct 20 2001
Sum of n terms of an arithmetic progression with the first term 1 and the common difference n: a(1)=1, a(2) = 1+3, a(3) = 1+4+7, a(4) = 1+5+9+13, etc. - Amarnath Murthy, Mar 25 2004
This is identical to: first triangular number A000217, 2nd square number A000290, 3rd pentagonal number A000326, 4th hexagonal number A000384, 5th heptagonal number A000566, 6th octagonal number A000567, ..., (n+1)-th (n+3)-gonal number = main diagonal of rectangular array T(n,k) of polygonal numbers, by diagonals, referred to in A086271. - Jonathan Vos Post, Dec 19 2007
Also (n + 1)! times the determinant of the n X n matrix given by m(i,j) = (i+1)/i if i=j and otherwise 1. For example, (6 + 1)!*Det[{{2,1,1,1,1,1}, {1,3/2,1,1,1,1},{1,1,4/3,1,1,1}, {1,1,1,5/4,1,1}, {1,1,1,1,6/5,1}, {1,1,1,1,1,7/6}}] = 154 = a(6). - John M. Campbell, May 20 2011
a(n-1) = N_2(n), n>=1, is the number of 2-faces of n planes in generic position in three-dimensional space. See comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - Wolfdieter Lang, May 27 2011
For n>2, a(n) is 2 * (average cycle weight of primitive Hamiltonian cycles on a simply weighted K_n) (see link). - Jon Perry, Nov 23 2014
a(n) is the partial sums of A104249. - J. M. Bergot, Dec 28 2014
Sum of the numbers in the 1st column of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021
From Enrique Navarrete, Mar 27 2023: (Start)
a(n) is the number of ordered set partitions of an (n+1)-set into 2 sets such that the first set has 0, 1, or 2 elements, the second set has no restrictions, and we choose an element from the second set. For n=4, the a(4) = 55 set partitions of [5] are the following (where the element selected from the second set is in parentheses):
   { }, {(1), 2, 3, 4, 5}  (5 of these);
   {1}, {(2), 3, 4, 5}   (20 of these);
   {1, 2}, {(3), 4, 5}   (30 of these). (End)

V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

William A. Tedeschi, Table of n, a(n) for n = 0..10000
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-382. [See p. 301].
Jon Perry, Weighted Hamiltonian Cycles
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Eric Weisstein's World of Mathematics, Polygonal Number
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000124.

A006000:=(1+2*z**2)/(z-1)**4; # Simon Plouffe in his 1992 dissertation

a[n_]:=(n^3-n^2)/2+n; Table[a[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
CoefficientList[Series[(1 + 2 x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Nov 21 2014 *)

def a(n):
    return n + (n**3 - n**2)//2 # William A. Tedeschi, Aug 22 2010

a(n) = Sum_{j=1..n+1} (binomial(0,0*j) + binomial(n+1,2)). - Zerinvary Lajos, Jul 25 2006
a(n-1) = n + (n^3 - n^2)/2 = n + n*T(n-1) where T(n-1) is a triangular number, n >= 1. - William A. Tedeschi, Aug 22 2010
a(n) = A002817(n)*4/n for n > 0. - Jon Perry, Nov 21 2014
E.g.f.: (1 + x)*(2 + 4*x + x^2)*exp(x)/2. - Robert Israel, Nov 24 2014
a(n) = A057145(n+3,n+1). - R. J. Mathar, Jul 28 2016
a(n) = A000124(n) * (n+1). - Alois P. Heinz, Aug 31 2023

A161706 a(n) = (-11n^5 + 145n^4 - 635n^3 + 1115n^2 - 494*n + 120)/120.

Original entry on oeis.org

1, 2, 4, 5, 10, 20, 21, -27, -201, -626, -1486, -3035, -5608, -9632, -15637, -24267, -36291, -52614, -74288, -102523, -138698, -184372, -241295, -311419, -396909, -500154, -623778, -770651, -943900, -1146920, -1383385, -1657259, -1972807

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 20:

a(n) = A027750(A006218(19) + k + 1), 0 <= k < A000005(20).

			Differences of divisors of 20 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     4     5    10    20
     1     2     1     5    10
        1    -1     4     5
          -2     5     1
              7    -4
               -11

G. C. Greubel, Table of n, a(n) for n = 0..1000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.

Cf. A005018, A161700, A161856. - Reinhard Zumkeller, Jun 21 2009

[(-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161706:=n->(-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120: seq(A161706(n), n=0..50); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1 - 4*x + 7*x^2 - 9*x^3 + 15*x^4 - 21*x^5)/(1 - x)^6, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

a(n)=(-11*n^5+145*n^4-635*n^3+1115*n^2-494*n+120)/120 \\ Charles R Greathouse IV, Sep 24 2015

def A161706(n): return (n*(n*(n*(n*(145 - 11*n) - 635) + 1115) - 494) + 120)//15>>3 # Chai Wah Wu, Oct 23 2023

a(n) = C(n,0) + C(n,1) + C(n,2) - 2*C(n,3) + 7*C(n,4) - 11*C(n,5).

G.f.: (1-4*x+7*x^2-9*x^3+15*x^4-21*x^5)/(1-x)^6. - Colin Barker, Apr 25 2012

A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 24, 39, -2, -295, -1308, -3980, -9996, -22150, -44808, -84483, -150534, -256001, -418588, -661806, -1016288, -1521288, -2226376, -3193341, -4498314, -6234123, -8512892, -11468896, -15261684, -20079482, -26142888

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 24:

a(n) = A027750(A006218(23) + k + 1), 0 <= k < A000005(24).

			Differences of divisors of 24 to compute the coefficients of their interpolating polynomial, see formula:
1 ... 2 ... 3 ... 4 ... 6 ... 8 .. 12 .. 24
.. 1 ... 1 ... 1 ... 2 ... 2 ... 4 .. 12
..... 0 ... 0 ... 1 ... 0 ... 2 ... 8
........ 0 ... 1 .. -1 ... 2 ... 6
........... 1 .. -2 ... 3 ... 4
............. -3 ... 5 ... 1
................. 8 .. -4
.................. -12.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A080856, A161711, A161712, A161713, A161715, A006261, A018253, A161700, A161856.

[(-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^ 3 + 22561*n^2 - 7302*n + 2520)/2520: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(-6n^7+154n^6-1533n^5+7525n^4-18879n^3+22561n^2-7302n+2520)/2520,{n,0,40}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,2,3,4,6,8,12,24},40] (* Harvey P. Dale, Jul 15 2012 *)

a(n)=(-6*n^7+154*n^6-1533*n^5+7525*n^4-18879*n^3+22561*n^2-7302*n+2520)/2520 \\ Charles R Greathouse IV, Sep 24 2015

A161710_list, m = [1], [-12, 80, -223, 333, -281, 127, -23, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,4) - 3*C(n,5) + 8*C(n,6) - 12*C(n,7).
G.f.: (1-6*x+15*x^2-20*x^3+16*x^4-12*x^5+18*x^6-24*x^7)/(1-x)^8. - Bruno Berselli, Jul 17 2011
a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=6, a(5)=8, a(6)=12, a(7)=24, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+ 8*a(n-7)- a(n-8). - Harvey P. Dale, Jul 15 2012

A161713 a(n) = (-n^5 + 15n^4 - 65n^3 + 125n^2 - 34n + 40)/40.

Original entry on oeis.org

1, 2, 4, 7, 14, 28, 49, 71, 79, 46, -70, -329, -812, -1624, -2897, -4793, -7507, -11270, -16352, -23065, -31766, -42860, -56803, -74105, -95333, -121114, -152138, -189161, -233008, -284576, -344837, -414841, -495719, -588686, -695044

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 28:

a(n) = A027750(A006218(27) + k + 1), 0 <= k < A000005(28).

			Differences of divisors of 28 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     4     7    14    28
     1     2     3     7    14
        1     1     4     7
           0     3     3
              3     0
                -3

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A018254, A058331, A080856, A086514, A161700, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161715, A161856.

[(-n^5 + 15*n^4 - 65*n^3 + 125*n^2 - 34*n + 40)/40: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(-n^5+15n^4-65n^3+125n^2-34n)/40+1,{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,4,7,14,28},40] (* Harvey P. Dale, Jan 14 2014 *)

a(n)=(-n^5+15*n^4-65*n^3+125*n^2-34*n+40)/40 \\ Charles R Greathouse IV, Sep 24 2015

def A161713(n): return n*(n*(n*(n*(15 - n) - 65) + 125) - 34)//40 + 1 # Chai Wah Wu, Dec 16 2021

a(n) = C(n,0) + C(n,1) + C(n,2) + 3*C(n,4) - 3*C(n,5).
G.f.: -(-1+4*x-7*x^2+7*x^3-7*x^4+7*x^5)/(-1+x)^6. - R. J. Mathar, Jun 18 2009
a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=14, a(5)=28, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jan 14 2014

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 15, 30, 171, 886, 3359, 10143, 26072, 59502, 123931, 240048, 438261, 761754, 1270123, 2043641, 3188202, 4840994, 7176951, 10416034, 14831391, 20758446, 28604967, 38862163, 52116860, 69064806, 90525155, 117456180

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 30:

a(n) = A027750(A006218(29) + k + 1), 0 <= k < A000005(30).

			Differences of divisors of 30 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     5     6    10    15    30
     1     1     2     1     4     5    15
        0     1    -1     3     1    10
           1    -2     4    -2     9
             -3     6    -6    11
                 9   -12    17
                  -21    29
                      50

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713.

Cf. A018255, A161700, A161856. - Reinhard Zumkeller, Jun 21 2009

[(50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

CoefficientList[Series[(1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

x='x+O('x^50); Vec((1 -6*x +15*x^2 -19*x^3 +8*x^4 +18*x^5 -51*x^6 +84*x^7) /(-1+x)^8) \\ G. C. Greubel, Jul 16 2017

A161710_list, m = [1], [50, -321, 864, -1249, 1024, -452, 85, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,3) - 3*C(n,4) + 9*C(n,5) - 21*C(n,6) + 50*C(n,7).
G.f.: (1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8. - R. J. Mathar, Jun 18 2009
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Apr 26 2021

A161701 a(n) = (n^5 - 5n^4 + 5n^3 + 5n^2 + 114n + 120)/120.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 28, 64, 135, 262, 473, 804, 1300, 2016, 3018, 4384, 6205, 8586, 11647, 15524, 20370, 26356, 33672, 42528, 53155, 65806, 80757, 98308, 118784, 142536, 169942, 201408, 237369, 278290, 324667, 377028, 435934, 501980, 575796, 658048

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 12:

a(n) = A027750(A006218(11) + k + 1), 0 <= k < A000005(12).

			Differences of divisors of 12 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     4     6    12
     1     1     1     2     6
        0     0     1     4
           0     1     3
              1     2
                 1

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715.

[(n^5 - 5*n^4 + 5*n^3 + 5*n^2 + 114*n + 120)/120: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161701:=n->(n^5 - 5*n^4 + 5*n^3 + 5*n^2 + 114*n + 120)/120: seq(A161701(n), n=0..60); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1-4*x+6*x^2-4*x^3+2*x^4)/(1-x)^6, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

a(n)=(n^5-5*n^4+5*n^3+5*n^2+114*n+120)/120 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + C(n,1) + C(n,4) + C(n,5).

G.f.: (1-4*x+6*x^2-4*x^3+2*x^4)/(1-x)^6. - Colin Barker, Aug 20 2012

A161704 a(n) = (3n^5 - 35n^4 + 145n^3 - 235n^2 + 152*n + 30)/30.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 59, 190, 513, 1186, 2435, 4566, 7977, 13170, 20763, 31502, 46273, 66114, 92227, 125990, 168969, 222930, 289851, 371934, 471617, 591586, 734787, 904438, 1104041, 1337394, 1608603, 1922094, 2282625, 2695298, 3165571, 3699270

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 6} = divisors of 18:

a(n) = A027750(A006218(17) + k + 1), 0 <= k < A000005(18).

			Differences of divisors of 18 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     6     9    18
     1     1     3     3     9
        0     2     0     6
           2    -2     6
             -4     8
                12

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A018251, A058331, A080856, A086514, A161701, A161702, A161703, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A161856.

[(3*n^5 - 35*n^4 + 145*n^3 - 235*n^2 + 152*n + 30)/30: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161704:=n->(3*n^5 - 35*n^4 + 145*n^3 - 235*n^2 + 152*n + 30)/30: seq(A161704(n), n=0..50); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1 - 4*x + 6*x^2 - 2*x^3 - 7*x^4 + 18*x^5)/(x - 1)^6, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

a(n)=n*(3*n^4-35*n^3+145*n^2-235*n+152)/30+1

a(n) = C(n,0) + C(n,1) + 2*C(n,3) - 4*C(n,4) + 12*C(n,5).

G.f.: ( 1-4*x+6*x^2-2*x^3-7*x^4+18*x^5 ) / (x-1)^6. - R. J. Mathar, Jul 12 2016

A161708 a(n) = -n^3 + 7n^2 - 5n + 1.

Original entry on oeis.org

1, 2, 11, 22, 29, 26, 7, -34, -103, -206, -349, -538, -779, -1078, -1441, -1874, -2383, -2974, -3653, -4426, -5299, -6278, -7369, -8578, -9911, -11374, -12973, -14714, -16603, -18646, -20849, -23218, -25759, -28478, -31381, -34474, -37763, -41254

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 22:

a(n) = A027750(A006218(21) + k + 1), 0 <= k < A000005(22).

			Differences of divisors of 22 to compute the coefficients of their interpolating polynomial, see formula:
  1     2    11    22
     1     9    11
        8     2
          -6

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161710, A080856, A161711, A161712, A161713, A161715, A006261.

[-n^3 + 7*n^2 - 5*n + 1: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[-n^3+7n^2-5n+1,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,2,11,22},40] (* Harvey P. Dale, Nov 12 2013 *)

a(n)=-n^3+7*n^2-5*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + C(n,1) + 8*C(n,2) - 6*C(n,3).
G.f.: -(-1+2*x-9*x^2+14*x^3)/(-1+x)^4. - R. J. Mathar, Jun 18 2009
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) with a(0)=1, a(1)=2, a(2)=11, a(3)=22. - Harvey P. Dale, Nov 12 2013
E.g.f.: (-x^3 + 4*x^2 + x + 1)*exp(x). - G. C. Greubel, Jul 16 2017

cp's OEIS Frontend

A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.

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A086514 Difference between the arithmetic mean of the neighbors of the terms and the term itself follows the pattern 0,1,2,3,4,5,...

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A006000 a(n) = (n+1)*(n^2+n+2)/2; g.f.: (1 + 2*x^2) / (1 - x)^4.

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A161706 a(n) = (-11*n^5 + 145*n^4 - 635*n^3 + 1115*n^2 - 494*n + 120)/120.

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A161710 a(n) = (-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^3 + 22561*n^2 - 7302*n + 2520)/2520.

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A161713 a(n) = (-n^5 + 15*n^4 - 65*n^3 + 125*n^2 - 34*n + 40)/40.

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A006000 a(n) = (n+1)(n^2+n+2)/2; g.f.: (1 + 2x^2) / (1 - x)^4.

A161706 a(n) = (-11n^5 + 145n^4 - 635n^3 + 1115n^2 - 494*n + 120)/120.

A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.

A161713 a(n) = (-n^5 + 15n^4 - 65n^3 + 125n^2 - 34n + 40)/40.

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.

A161701 a(n) = (n^5 - 5n^4 + 5n^3 + 5n^2 + 114n + 120)/120.

A161704 a(n) = (3n^5 - 35n^4 + 145n^3 - 235n^2 + 152*n + 30)/30.

A161708 a(n) = -n^3 + 7n^2 - 5n + 1.