A161708 -id:A161708 - cp's OEIS frontend

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

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

A161702 a(n) = (-n^3 + 9n^2 - 5n + 3)/3.

Original entry on oeis.org

1, 2, 7, 14, 21, 26, 27, 22, 9, -14, -49, -98, -163, -246, -349, -474, -623, -798, -1001, -1234, -1499, -1798, -2133, -2506, -2919, -3374, -3873, -4418, -5011, -5654, -6349, -7098, -7903, -8766, -9689, -10674, -11723, -12838, -14021, -15274

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 14:

a(n) = A027750(A006218(13) + k + 1), 0 <= k < A000005(14).

			Differences of divisors of 14 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     7    14
     1     5     7
        4     2
          -2

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 (4,-6,4,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161703, A161704, A161706-A161708, A161710, A161711-A161713, A161715.

[(-n^3 + 9*n^2 - 5*n + 3)/3: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161702:=n->(-n^3 + 9*n^2 - 5*n + 3)/3: seq(A161702(n), n=0..60); # Wesley Ivan Hurt, Jul 16 2017

Table[(-n^3+9n^2-5n+3)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,2,7,14},40] (* Harvey P. Dale, Jun 15 2013 *)

a(n)=(-n^3+9*n^2-5*n+3)/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + C(n,1) + 4*C(n,2) - 2*C(n,3).
G.f.: (1-2*x+5*x^2-6*x^3)/(1-x)^4. - Colin Barker, Jan 08 2012
a(0)=1, a(1)=2, a(2)=7, a(3)=14, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jun 15 2013

A161703 a(n) = (4n^3 - 12n^2 + 14*n + 3)/3.

Original entry on oeis.org

1, 3, 5, 15, 41, 91, 173, 295, 465, 691, 981, 1343, 1785, 2315, 2941, 3671, 4513, 5475, 6565, 7791, 9161, 10683, 12365, 14215, 16241, 18451, 20853, 23455, 26265, 29291, 32541, 36023, 39745, 43715, 47941, 52431, 57193, 62235, 67565, 73191, 79121

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 15:

a(n) = A027750(A006218(14) + k + 1), 0 <= k < A000005(15).

			Differences of divisors of 15 to compute the coefficients of their interpolating polynomial, see formula:
  1     3     5    15
     2     2    10
        0     8
           8

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 (4,-6,4,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161704, A161706-A161708, A161710, A161711-A161713, A161715.

[(4*n^3 - 12*n^2 + 14*n + 3)/3: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161703:=n->(4*n^3 - 12*n^2 + 14*n + 3)/3: seq(A161703(n), n=0..100); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1 - x - x^2 + 9*x^3)/(1 - x)^4, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)
LinearRecurrence[{4,-6,4,-1},{1,3,5,15},50] (* Harvey P. Dale, Oct 04 2024 *)

a(n)=n*(4*n^2-12*n+14)/3+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + 2*C(n,1) + 8*C(n,3).

G.f.: (1-x-x^2+9*x^3)/(1-x)^4. - Colin Barker, Jan 08 2012

A161711 a(n) = (-4n^3 + 27n^2 - 20*n + 3)/3.

Original entry on oeis.org

1, 2, 13, 26, 33, 26, -3, -62, -159, -302, -499, -758, -1087, -1494, -1987, -2574, -3263, -4062, -4979, -6022, -7199, -8518, -9987, -11614, -13407, -15374, -17523, -19862, -22399, -25142, -28099, -31278, -34687, -38334, -42227, -46374, -50783

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 26:

a(n) = A027750(A006218(25) + k + 1), 0 <= k < A000005(26).

			Differences of divisors of 26 to compute the coefficients of their interpolating polynomial, see formula:
  1     2    13    26
     1    11    13
       10     2
          -8

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 (4,-6,4,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161712, A161713, A161715, A006261.

[(-4*n^3 + 27*n^2 - 20*n + 3)/3: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

LinearRecurrence[{4,-6,4,-1},{1,2,13,26},40] (* Harvey P. Dale, Jul 02 2017 *)

x='x+O('x^50); Vec((1-2*x+11*x^2-18*x^3)/(1-x)^4) \\ G. C. Greubel, Jul 16 2017

a(n) = C(n,0) + C(n,1) + 10*C(n,2) - 8*C(n,3).

G.f.: (1-2*x+11*x^2-18*x^3)/(1-x)^4. - Bruno Berselli, Jul 17 2011

A161856 Triangle read by rows in which row n lists the coefficients of the interpolating polynomial for its divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 0, 2, 1, 6, 1, 1, 1, 1, 1, 2, 4, 1, 1, 2, 0, 1, 10, 1, 1, 0, 0, 1, 1, 1, 12, 1, 1, 4, -2, 1, 2, 0, 8, 1, 1, 1, 1, 1, 1, 16, 1, 1, 0, 2, -4, 12, 1, 18, 1, 1, 1, -2, 7, -11, 1, 2, 2, 8, 1, 1, 8, -6, 1, 22, 1, 1, 0, 0, 1, -3, 8, -12, 1, 4, 16, 1, 1, 10, -8, 1, 2, 4, 8, 1, 1

Offset: 1

Reinhard Zumkeller, Jun 20 2009

EDP(n,x) = SUM(a(A006218(n)-1+i)*A007318(x,i-1): 1<=i<=A000005(n)) is the interpolating polynomial for the divisors of n, see also A161700;
A000005(n) = length of n-th row, i.e. same length as n-th row in A027750;
sum of n-th row, n>1: A161857(n) = SUM(a(A006218(n-1)+i): 1<=i<=A000005(n));
a(A006218(n)+1) = 1.

			1; 1,1; 1,2; 1,1,1; 1,4; 1,1,0,2; 1,6; 1,1,1,1; 1,2,4; ... .

R. Zumkeller, Table of rows 1..1000, flattened
R. Zumkeller, Enumerations of Divisors

A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A006261, A161715.

cp's OEIS Frontend

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

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

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

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A161702 a(n) = (-n^3 + 9n^2 - 5n + 3)/3.

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Programs

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.

A161703 a(n) = (4n^3 - 12n^2 + 14*n + 3)/3.

A161711 a(n) = (-4n^3 + 27n^2 - 20*n + 3)/3.