A140107 -id:A140107 - cp's OEIS frontend

A128965 a(n) = (n^3 - n)*7^n.

0, 294, 8232, 144060, 2016840, 24706290, 276710448, 2905459704, 29054597040, 279650496510, 2610071300760, 23751648836916, 211605598728888, 1851548988877770, 15951806673408480, 135590356723972080, 1138958996481365472, 9467596658251350486, 77968443067952298120

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 (28,-294,1372,-2401).

Cf. A000420, A007531, A036289, A128796, A140107.

Cf. A128960, A128961, A128962, A128963, A128964, A128967, A128969.

[(n^3 - n)*7^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{28, -294, 1372, -2401}, {0, 294, 8232, 144060}, 30] (* Vincenzo Librandi, Feb 11 2013 *)
Table[(n^3-n)7^n,{n,20}] (* Harvey P. Dale, May 14 2020 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 294x^2/(1-7x)^4.
a(n) = 294*A140107(n-2). (End)
a(n) = 28*a(n-1) - 294*a(n-2) + 1372*a(n-3) - 2401*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000420(n).
Sum_{n>=2} 1/a(n) = (18/7)*log(7/6) - 11/28.
Sum_{n>=2} (-1)^n/a(n) = (32/7)*log(8/7) - 17/28. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A218017 Triangle, read by rows, where T(n,k) = k!C(n, k)7^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 7, 1, 49, 14, 2, 343, 147, 42, 6, 2401, 1372, 588, 168, 24, 16807, 12005, 6860, 2940, 840, 120, 117649, 100842, 72030, 41160, 17640, 5040, 720, 823543, 823543, 705894, 504210, 288120, 123480, 35280, 5040, 5764801, 6588344, 6588344, 5647152, 4033680, 2304960, 987840, 282240, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=7. Also:
first column:    A000420;
second column:   A027473;
third column:  2*A027474;
fourth column: 6*A140107.

			Triangle begins:
1;
7,       1;
49,      14,      2;
343,     147,     42,      6;
2401,    1372,    588,     168,     24;
16807,   12005,   6860,    2940,    840,     120;
117649,  100842,  72030,   41160,   17640,   5040,    720;
823543,  823543,  705894,  504210,  288120,  123480,  35280,  5040; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000420, A027466, A027473, A027474, A090802, A140107, A217629, A218016.

[Factorial(n)/Factorial(n-k)*7^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*7^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 7^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(7x)*x^k.

A170932 a(n) = binomial(n + 8, 8)*7^n .

Original entry on oeis.org

1, 63, 2205, 56595, 1188495, 21630609, 353299947, 5299499205, 74192988870, 980996186170, 12360551945742, 149450309889426, 1743586948709970, 19715944727720430, 216875392004924730, 2327795874186192102, 24441856678955017071, 251607348165713411025

Offset: 0

Zerinvary Lajos, Feb 08 2010

With a different offset, number of n-permutations of 8 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight, (8) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (63,-1764,28812,-302526,2117682,-9882516,29647548,-51883209,40353607).

Cf. A027474, A140107, A139641, A140404, A036226, A050989.

[Binomial(n + 8, 8)*7^n: n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 8, 8]*7^n, {n, 0, 20}]

a(n) = C(n + 8, 8)*7^n.
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 12082656/5 - 15676416*log(7/6).
Sum_{n>=0} (-1)^n/a(n) = 117440512*log(8/7) - 235229912/15. (End)

A317014 Triangle read by rows: T(0,0) = 1; T(n,k) = 7 * T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2). T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 7, 49, 1, 343, 14, 2401, 147, 1, 16807, 1372, 21, 117649, 12005, 294, 1, 823543, 100842, 3430, 28, 5764801, 823543, 36015, 490, 1, 40353607, 6588344, 352947, 6860, 35, 282475249, 51883209, 3294172, 84035, 735, 1, 1977326743, 403536070, 29647548, 941192, 12005, 42

Offset: 0

Zagros Lalo, Jul 19 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013614 ((1+7*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A027466 ((7+x)^n).
The coefficients in the expansion of 1/(1-7x-x^2) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 7.14005494464025913554... ((7+sqrt(53))/2), a metallic mean (see A176439), when n approaches infinity.

			Triangle begins:
1;
7;
49, 1;
343, 14;
2401, 147, 1;
16807, 1372, 21;
117649, 12005, 294, 1;
823543, 100842, 3430, 28;
5764801, 823543, 36015, 490, 1;
40353607, 6588344, 352947, 6860, 35;
282475249, 51883209, 3294172, 84035, 735, 1;
1977326743, 403536070, 29647548, 941192, 12005, 42;
13841287201, 3107227739, 259416045, 9882516, 168070, 1029, 1;
96889010407, 23727920916, 2219448385, 98825160, 2117682, 19208, 49;
678223072849, 179936733613, 18643366434, 951192165, 24706290, 302526, 1372, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 96.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 7x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (7 + x)^n

Row sums give A054413.
Cf. A013614, A027466, A176439.
Cf. A000420 (column 0), A027473 (column 1), A027474 (column 2), A140107 (column 3), A139641 (column 4).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 7 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 7*T(n-1, k)+T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

A197192 a(n) = binomial(n+9, 9)*7^n.

Original entry on oeis.org

1, 70, 2695, 75460, 1716715, 33647614, 588833245, 9421331920, 140142312310, 1961992372340, 26094498552122, 332111799754280, 4068369546989930, 48194531556649940, 554237112901474310, 6207455664496512272, 67894046330430602975

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A139641, A140107, A140404, A170932

Magma
```
[Binomial(n+9, 9)*7^n: n in [0..20]];
```

Table[Binomial[n+9,9]7^n,{n,0,20}] (* Harvey P. Dale, Jul 10 2025 *)

a(n) = C(n + 9, 9)*7^n.

A197193 a(n) = binomial(n+10, 10)*7^n.

Original entry on oeis.org

1, 77, 3234, 98098, 2403401, 50471421, 942133192, 16016264264, 252256162158, 3727785507446, 52188997104244, 697434779483988, 8950413003377846, 110847422580294862, 1330169070963538344, 15518639161241280680, 176524520459119567735, 1962537315692564605995, 21369850770874592376390, 228319984551975908021430

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (77,-2695,56595,-792330,7764834,-54353838,271769190,-951192165,2219448385,-3107227739,1977326743).

Cf. A139641, A140107, A140404, A170932

Magma
```
[Binomial(n+10, 10)*7^n: n in [0..20]];
```

Table[Binomial[n+10,10]7^n,{n,0,30}] (* or *) LinearRecurrence[{77,-2695,56595,-792330,7764834,-54353838,271769190,-951192165,2219448385,-3107227739,1977326743},{1,77,3234,98098,2403401,50471421,942133192,16016264264,252256162158,3727785507446,52188997104244},30] (* Harvey P. Dale, Jul 11 2025 *)

a(n) = C(n + 10, 10)*7^n.

G.f.: -1 / (7*x-1)^11 . - R. J. Mathar, Oct 13 2011

cp's OEIS Frontend

A128965 a(n) = (n^3 - n)*7^n.

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A218017 Triangle, read by rows, where T(n,k) = k!*C(n, k)*7^(n-k) for n>=0, k=0..n.

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A170932 a(n) = binomial(n + 8, 8)*7^n .

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A317014 Triangle read by rows: T(0,0) = 1; T(n,k) = 7 * T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2). T(n,k)=0 for n or k < 0.

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A197192 a(n) = binomial(n+9, 9)*7^n.

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Programs

Magma

Mathematica

Formula

A197193 a(n) = binomial(n+10, 10)*7^n.

Views

Author

Keywords

Links

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Programs

Magma

Mathematica

Formula

A218017 Triangle, read by rows, where T(n,k) = k!C(n, k)7^(n-k) for n>=0, k=0..n.