A016181 -id:A016181 - cp's OEIS frontend

A327317 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 2 and s = 1/2.

1, 5, 4, 21, 30, 12, 85, 168, 120, 32, 341, 850, 840, 400, 80, 1365, 4092, 5100, 3360, 1200, 192, 5461, 19110, 28644, 23800, 11760, 3360, 448, 21845, 87376, 152880, 152768, 95200, 37632, 8960, 1024, 87381, 393210, 786384, 917280, 687456, 342720, 112896

Offset: 1

Clark Kimberling, Nov 03 2019

p(x,n) is a strong divisibility sequence of polynomials. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			First six rows:
     1;
     5,    4;
    21,   30,   12;
    85,  168,  120,   32;
   341,  850,  840,  400,   80;
  1365, 4092, 5100, 3360, 1200, 192;
The first six polynomials, not factored:
1, 5 + 4 x, 21 + 30 x + 12 x^2, 85 + 168 x + 120 x^2 + 32 x^3, 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 1365 + 4092 x + 5100 x^2 + 3360 x^3 + 1200 x^4 + 192 x^5.
The first six polynomials, factored:
1, 5 + 4 x, 3 (7 + 10 x + 4 x^2), (5 + 4 x) (17 + 20 x + 8 x^2), 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 3 (5 + 4 x) (7 + 10 x + 4 x^2) (13 + 10 x + 4 x^2).

Cf. A327316, A002450 (x=0), A016137 (x=1), A001045 (x = -1), A016162 (x = 2), A016181 (x = 3), A016127 (x = -3), A016157 (x = 1/2).

r = 2; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327317 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327317 sequence *)

A248341 a(n) = 10^n - 7^n.

Original entry on oeis.org

0, 3, 51, 657, 7599, 83193, 882351, 9176457, 94235199, 959646393, 9717524751, 98022673257, 986158712799, 9903110989593, 99321776927151, 995252438490057, 9966767069430399, 99767369486012793, 998371586402089551, 9988601104814626857

Offset: 0

Vincenzo Librandi, Oct 05 2014

Index entries for linear recurrences with constant coefficients, signature (17,-70).

Cf. sequences of the form k^n-7^n: A016177 (k=8), A191467 (k=9), this sequence(k=10), A139745 (k=11).

Cf. A000420, A011557, A016181.

Magma
```
[10^n-7^n: n in [0..30]];
```

Table[10^n - 7^n, {n, 0, 25}] (* or *) CoefficientList[Series[3 x/((1 - 7 x) (1 - 10 x)), {x, 0, 30}], x]
LinearRecurrence[{17,-70},{0,3},20] (* Harvey P. Dale, Dec 18 2020 *)

G.f.: 3*x/((1-7*x)*(1-10*x)).
a(n) = 17*a(n-1) - 70*a(n-2).
a(n) = A011557(n) - A000420(n).
a(n+1) = 3*A016181(n).
E.g.f.: 2*exp(17*x/2)*sinh(3*x/2). - Elmo R. Oliveira, Apr 01 2025

A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 5, 9, 5, 0, 1, 7, 21, 27, 11, 0, 1, 9, 39, 85, 81, 21, 0, 1, 11, 63, 203, 341, 243, 43, 0, 1, 13, 93, 405, 1031, 1365, 729, 85, 0, 1, 15, 129, 715, 2511, 5187, 5461, 2187, 171, 0, 1, 17, 171, 1157, 5261, 15309, 25999, 21845, 6561, 341, 0, 1

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 09 2005

Consider a 3 X 3 matrix M =
  [n, 1, 1]
  [1, n, 1]
  [1, 1, n].
The n-th row of the array contains the values of the nondiagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = nondiagonal entry + (n-1)^k.)
Table:
  n: row sequence G.f. cross references.
  0: (2^n-(-1)^n)/3 1/((1+1x)(1-2x)) A001045 (Jacobsthal sequence)
  1: (3^n-0^n)/3 1/(1-3x) A000244
  2: (4^n-1^n)/3 1/((1-1x)(1-4x)) A002450
  3: (5^n-2^n)/3 1/((1-2x)(1-5x)) A016127
  4: (6^n-3^n)/3 1/((1-3x)(1-6x)) A016137
  5: (7^n-4^n)/3 1/((1-4x)(1-7x)) A016150
  6: (8^n-5^n)/3 1/((1-5x)(1-8x)) A016162
  7: (9^n-6^n)/3 1/((1-6x)(1-9x)) A016172
  8: (10^n-7^n)/3 1/((1-7x)(1-10x)) A016181
  9: (11^n-8^n)/3 1/((1-8x)(1-11x)) A016187
  10:(12^n-9^n)/3 1/((1-9x)(1-12x)) A016191
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+2.
Columns follow polynomials with certain binomial coefficients:
  column: polynomial
  0: 0
  1: 1
  2: 2*n + 1
  3: 3*n^2+ 3*n + 3
  4: 4*n^3+ 6*n^2+ 12*n + 5
  5: 5*n^4+10*n^3+ 30*n^2+ 25*n + 11
  6: 6*n^5+15*n^4+ 60*n^3+ 75*n^2+ 66*n + 21
  7: 7*n^6+21*n^5+105*n^4+ 175*n^3+ 231*n^2+ 147*n + 43
  8: 8*n^7+28*n^6+168*n^5+ 350*n^4+ 616*n^3+ 588*n^2+344*n+ 85
  etc.
Coefficients are generated by the array T(n,k)=(2^(n-k-1)-(-1)^(n-k-1))/3*(binomial(k+(n-k-1),n-k-1)) read by antidiagonals.

			Array begins:
  0, 1, 1,  3,   5,   11, ...
  0, 1, 3,  9,  27,   81, ...
  0, 1, 5, 21,  85,  341, ...
  0, 1, 7, 39, 203, 1031, ...
  0, 1, 9, 63, 405, 2511, ...
  ...

MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M for(k=0,10, for(i=0,10,print1((MM(3,k)^i)[1,2],","));print())

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A327317 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 2 and s = 1/2.

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A248341 a(n) = 10^n - 7^n.

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A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

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