A168635 -id:A168635 - cp's OEIS frontend

A168067 a(n) = n*(n^7+1)/2.

0, 1, 129, 3282, 32770, 195315, 839811, 2882404, 8388612, 21523365, 50000005, 107179446, 214990854, 407865367, 737894535, 1281445320, 2147483656, 3487878729, 5509980297, 8491781530, 12800000010, 18911429691, 27437936779, 39155492652, 55037657100, 76293945325

Offset: 0

N. J. A. Sloane, Dec 11 2009

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)

Cf. A168635.

[n*(n^7+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 10 2014

Table[n (n^7 + 1)/2, {n, 0, 40}] (* or *) CoefficientList[Series[x (1 + 120 x + 2157 x^2 + 7792 x^3 + 7827 x^4 + 2136 x^5 + 127 x^6) / (1 - x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2014 *)

G.f.: x*(1 + 120*x + 2157*x^2 + 7792*x^3 + 7827*x^4 + 2136*x^5 + 127*x^6)/(1-x)^9. - Vincenzo Librandi, Dec 10 2014

More terms from Vincenzo Librandi, Dec 10 2014

A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 12, 1, 4, 12, 36, 108, 1, 5, 20, 80, 320, 1280, 1, 6, 30, 150, 750, 3750, 18750, 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890

Offset: 0

Thomas Wieder, Mar 20 2009

Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).
Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.

			Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  2,   2,    2,     2,      2,       2,        2,        2, ... A040000;
  1,  3,   6,   12,    24,     48,      96,      192,      384, ... A003945;
  1,  4,  12,   36,   108,    324,     972,     2916,     8748, ... A003946;
  1,  5,  20,   80,   320,   1280,    5120,    20480,    81920, ... A003947;
  1,  6,  30,  150,   750,   3750,   18750,    93750,   468750, ... A003948;
  1,  7,  42,  252,  1512,   9072,   54432,   326592,  1959552, ... A003949;
  1,  8,  56,  392,  2744,  19208,  134456,   941192,  6588344, ... A003950;
  1,  9,  72,  576,  4608,  36864,  294912,  2359296, 18874368, ... A003951;
  1, 10,  90,  810,  7290,  65610,  590490,  5314410, 47829690, ... A003952;
  1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
  1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
  1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
  ... ;
The triangle begins as:
  1
  1, 1;
  1, 2,  2;
  1, 3,  6,  12;
  1, 4, 12,  36,  108;
  1, 5, 20,  80,  320,  1280;
  1, 6, 30, 150,  750,  3750,  18750;
  1, 7, 42, 252, 1512,  9072,  54432, 326592;
  1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
  ...;
T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Array rows n: A170733 (n=14), ..., A170769 (n=50).
Columns k: A000012(n) (k=0), A000027(n) (k=1), A002378(n-1) (k=2), A011379(n-1) (k=3), A179824(n) (k=4), A101362(n-1) (k=5), 2*A168351(n-1) (k=6), 2*A168526(n-1) (k=7), 2*A168635(n-1) (k=8), 2*A168675(n-1) (k=9), 2*A170783(n-1) (k=10), 2*A170793(n-1) (k=11).
Diagonals k: A055897 (k=n), A055541 (k=n-1), A373395 (k=n-2), A379612 (k=n-3).
Sums: (-1)^n*A065440(n) (signed row).
Cf. A000045, A000108, A000110, A007318, A079901, A158498.

A158497:= func< n,k | k le 1 select n^k else n*(n-1)^(k-1) >;
[A158497(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2025

A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
Table[A158497[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2025 *)

def A158497(n,k): return n^k if k<2 else n*(n-1)^(k-1)
print(flatten([[A158497(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 18 2025

T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.

Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.

Edited by R. J. Mathar, Mar 31 2009

More terms added by G. C. Greubel, Mar 18 2025

A168636 a(n) = n^7*(n^2 + 1)/2.

Original entry on oeis.org

0, 1, 320, 10935, 139264, 1015625, 5178816, 20588575, 68157440, 196101729, 505000000, 1188717431, 2597806080, 5333623945, 10383230144, 19307109375, 34493956096, 59499107585, 99485755200, 161790784759, 256640000000, 398040567561, 604881787840, 902278743455

Offset: 0

N. J. A. Sloane, Dec 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A168635.

[n^7*(n^2+1)/2: n in [0..25]]; // Vincenzo Librandi, Jul 29 2016

Table[n^7 (n^2+1)/2,{n,0,20}] (* or *) LinearRecurrence[{10,-45,120,-210, 252, -210, 120, -45,10,-1}, {0,1, 320, 10935, 139264, 1015625, 5178816, 20588575, 68157440, 196101729}, 21] (* Harvey P. Dale, Mar 09 2016 *)

a(n)=n^7*(n^2+1)/2 \\ Charles R Greathouse IV, Jul 29 2016

def A168636(n): return n^5*binomial(n^2+1,2)
print([A168636(n) for n in range(31)]) # G. C. Greubel, Mar 23 2025

From Harvey P. Dale, Mar 09 2016: (Start)
a(0)=0, a(1)=1, a(2)=320, a(3)=10935, a(4)=139264, a(5)=1015625, a(6)=5178816, a(7)=20588575, a(8)=68157440, a(9)=196101729, a(n)= 10*a(n-1)- 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10).
G.f.: x*(1 + 310*x + 7780*x^2 + 44194*x^3 + 76870*x^4 + 44194*x^5 + 7780*x^6 + 310*x^7 + x^8)/(1 - x)^10. (End)
E.g.f.: (1/2)*x*(2 + 318*x + 3326*x^2 + 8120*x^3 + 7091*x^4 + 2667*x^5 + 463*x^6 + 36*x^7 + x^8)*exp(x). - G. C. Greubel, Jul 28 2016

A168660 a(n) = n^7*(n^3 + 1)/2.

Original entry on oeis.org

0, 1, 576, 30618, 532480, 4921875, 30373056, 141649396, 537919488, 1745783685, 5005000000, 12978455886, 30976598016, 68960620183, 144680034240, 288410625000, 549890031616, 1008202119561, 1785539723328, 3065980064770

Offset: 0

N. J. A. Sloane, Dec 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A168635.

[n^7*(n^3+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Table[n^7*(n^3 + 1)/2, {n,0,50}] (* G. C. Greubel, Jul 28 2016 *)

a(n)=n^7*(n^3+1)/2 \\ Charles R Greathouse IV, Jul 29 2016

def A168660(n): return n^4*binomial(n^3+1,2)
print([A168660(n) for n in range(31)]) # G. C. Greubel, Mar 23 2025

From G. C. Greubel, Jul 28 2016: (Start)
G.f.: x*(1 + 565*x + 24337*x^2 + 227197*x^3 + 653875*x^4 + 656479*x^5 + 227995*x^6 + 23503*x^7 + 448*x^8)/(1 - x)^11.
E.g.f.: (1/2)*x*(2 + 574*x + 9631*x^2 + 34455*x^3 + 42665*x^4 + 22848*x^5 + 5881*x^6 + 750*x^7 + 45*x^8 + x^9)*exp(x). (End)

A168661 a(n) = n^7*(n^4 + 1)/2.

Original entry on oeis.org

0, 1, 1088, 89667, 2105344, 24453125, 181538496, 989075143, 4296015872, 15692921289, 50005000000, 142665578891, 371522101248, 896111571277, 2024835291584, 4324963359375, 8796227239936, 17136153323153, 32134511149632, 58245576384979, 102400640000000, 175139650815381, 292160397884608

Offset: 0

N. J. A. Sloane, Dec 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A168635.

[n^7*(n^4+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Table[n^7*(n^4 + 1)/2, {n,0,25}] (* G. C. Greubel, Jul 28 2016 *)

a(n)=n^7*(n^4+1)/2 \\ Charles R Greathouse IV, Jul 29 2016

def A168661(n): return n^3*binomial(n^4+1,2)
print([A168661(n) for n in range(31)]) # G. C. Greubel, Mar 24 2025

From G. C. Greubel, Jul 28 2016: (Start)
G.f.: x*(1 + 1076*x + 76677*x^2 + 1100928*x^3 + 4868154*x^4 + 7864728*x^5 + 4868154*x^6 + 1100928*x^7 + 76677*x^8 + 1076*x^9 + x^10)/(1 - x)^12.
E.g.f.: (1/2)*x*(2 + 1086*x + 28802*x^2 + 146100*x^3 + 246870*x^4 + 179508*x^5 + 63988*x^6 + 11880*x^7 + 1155*x^8 + 55*x^9 + x^10)*exp(x). (End)

A168662 a(n) = n^7*(n^5 + 1)/2.

Original entry on oeis.org

0, 1, 2112, 266814, 8396800, 122109375, 1088531136, 6921055372, 34360786944, 141217159725, 500005000000, 1569223931946, 4458068140032, 11649073935499, 28347008894400, 64873254375000, 140737622573056, 291311323784217, 578415996823104, 1106657906468950, 2048000640000000

Offset: 0

N. J. A. Sloane, Dec 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A168635.

[n^7*(n^5+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Table[n^7 (n^5+1)/2,{n,0,20}] (* Harvey P. Dale, May 17 2016 *)

a(n)=n^7*(n^5+1)/2 \\ Charles R Greathouse IV, Jul 29 2016

def A168662(n): return n^2*binomial(n^5+1,2)
print([A168662(n) for n in range(31)]) # G. C. Greubel, Mar 24 2025

From G. C. Greubel, Jul 28 2016: (Start)
G.f.: x*(1 + 2099*x + 239436*x^2 + 5092668*x^3 + 33159150*x^4 + 81259650*x^5 + 81252636*x^6 + 33159324*x^7 + 5095017*x^8 + 238835*x^9 + 1984*x^10)/(1-x)^13.
E.g.f.: (1/2)*x*(2 + 2110*x + 86827*x^2 + 611851*x^3 + 1379540*x^4 + 1323673*x^5 + 627397*x^6 + 159027*x^7 + 22275*x^8 + 1705*x^9 + 66*x^10 + x^11)*exp(x). (End)

cp's OEIS Frontend

A168067 a(n) = n*(n^7+1)/2.

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A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

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A168636 a(n) = n^7*(n^2 + 1)/2.

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A168660 a(n) = n^7*(n^3 + 1)/2.

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A168661 a(n) = n^7*(n^4 + 1)/2.

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A168662 a(n) = n^7*(n^5 + 1)/2.

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