A005062 -id:A005062 - cp's OEIS frontend

A016197 a(n) = 12^n - 11^n.

0, 1, 23, 397, 6095, 87781, 1214423, 16344637, 215622815, 2801832661, 35979939623, 457696700077, 5777672071535, 72470493235141, 904168630965623, 11229773405170717, 138934529031464255, 1713164078241143221

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (23, -132).

Cf. k^n-(k-1)^n: A000225 (k=2), A001047 (k=3), A005061 (k=4), A005060 (k=5), A005062 (k=6), A016169 (k=7), A016177 (k=8), A016185 (k=9), A016189 (k=10), A016195 (k=11), this sequence (k=12).

[12^n-11^n: n in [0..40]]; // Vincenzo Librandi, Aug 03 2014

f[n_]:= 12^n - 11^n; f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
CoefficientList[Series[x/((1 - 11 x) (1 - 12 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 03 2014 *)

a(n)=12^n-11^n \\ Charles R Greathouse IV, Aug 03 2014

G.f.: x/((1-11x)(1-12x)).
E.g.f.: e^(12*x)-e^(11*x). - Mohammad K. Azarian, Jan 14 2009
a(0)=0, a(n)=12*a(n-1)+11^(n-1). - _Vincenzo Librandi-, Feb 09 2011
a(0)=0, a(1)=1, a(n)=23*a(n-1)-132*a(n-2). - Vincenzo Librandi, Feb 09 2011

A327316 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = ((x+r)^n - (x+s)^n)/(r - s), where r = 3 and s = 2.

Original entry on oeis.org

1, 5, 2, 19, 15, 3, 65, 76, 30, 4, 211, 325, 190, 50, 5, 665, 1266, 975, 380, 75, 6, 2059, 4655, 4431, 2275, 665, 105, 7, 6305, 16472, 18620, 11816, 4550, 1064, 140, 8, 19171, 56745, 74124, 55860, 26586, 8190, 1596, 180, 9, 58025, 191710, 283725, 247080

Offset: 1

Clark Kimberling, Nov 01 2019

For every choice of integers r and s, the polynomials p(n,x) form a strong divisibility sequence. Thus, if r, s, and x are integers, then p(x,n) is a strong divisibility sequence. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)).

			First seven rows:
     1
     5      2
    19     15     3
    65     76    30     4
   211    325   190    50    5
   665   1266   975   380   75    6
  2059   4655  4431  2275  665  105   7

Cf. A001047 (x=0), A005061 (x=1), A005060 (x=2), A005062 (x=3), A081200 (x=1/2).

f[x_, n_] := ((x + r)^n - (x + s)^n)/(r - s);
r = 3; s = 2;
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}]] (* A327316 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327316 sequence *)

A343237 Triangle T obtained from the array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0, by reading antidiagonals upwards.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 15, 19, 7, 1, 1, 31, 65, 37, 9, 1, 1, 63, 211, 175, 61, 11, 1, 1, 127, 665, 781, 369, 91, 13, 1, 1, 255, 2059, 3367, 2101, 671, 127, 15, 1, 1, 511, 6305, 14197, 11529, 4651, 1105, 169, 17, 1

Offset: 0

Wolfdieter Lang, May 10 2021

This is the row reversed version of the triangle A047969(n, m). The corresponding array A047969 is a(n, k) = A(k, n) (transposed of array A).

A(n-1, k-1) = k^n - (k-1)^n gives the number of n-digit numbers with digits from K = {1, 2, 3, ..., k} such that any digit from K, say k, appears at least once. Motivated by a comment in A005061 by Enrique Navarrete, the instance k=4 for n >= 1 (the column 3 in array A), and the d = 3 (sub)-diagonal sequence of T for m >= 0.

			The array A begins:
n\k  0  1   2    3     4     5     6      7      8      9 ...
-------------------------------------------------------------
0:   1  1   1    1     1     1     1      1      1      1 ...
1:   1  3   5    7     9    11    13     15     17     19 ...
2:   1  7  19   37    61    91   127    169    217    271 ...
3:   1 15  65  175   369   671  1105   1695   2465   3439 ...
4:   1 31 211  781  2101  4651  9031  15961  26281  40951 ...
5:   1 63 665 3367 11529 31031 70993 144495 269297 468559 ...
...
The triangle T begins:
n\m   0    1     2     3     4     5    6    7   8  9 10 ...
-------------------------------------------------------------
0:    1
1:    1    1
2:    1    3     1
3:    1    7     5     1
4:    1   15    19     7     1
5:    1   31    65    37     9     1
6:    1   63   211   175    61    11    1
7:    1  127   665   781   369    91   13    1
8:    1  255  2059  3367  2101   671  127   15   1
9:    1  511  6305 14197 11529  4651 1105  169  17  1
10:   1 1023 19171 58975 61741 31031 9031 1695 217 19  1
...
Combinatorial interpretation (cf. A005061 by _Enrique Navarrete_)
The three digits numbers with digits from K ={1, 2, 3, 4} having at least one 4 are:
j=1 (one 4): 114, 141, 411; 224, 242, 422; 334, 343, 433; 124, 214, 142, 241, 412, 421; 134, 314, 143, 341, 413, 431; 234, 243, 423. That is,  3*3 + 3!*3 = 27 = binomial(3, 1)*(4-1)^(3-1) = 3*3^2;
j=2 (twice 4):  144, 414, 441;  244, 424, 442;  344, 434, 443; 3*3 = 9 = binomial(3, 2)*(4-1)^(3-2) = 3*3;
j=3 (thrice 4) 444; 1 = binomial(3, 3)*(4-1)^(3-3).
Together: 27 + 9 + 1 = 37 = A(2, 3) = T(5, 3).

Cf. A005061, A008292, A047969 (reversed), A045531 (central diagonal), A047970 (row sums of triangle).
Row sequences of array A (nexus numbers): A000012, A005408, A003215, A005917(k+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
Column sequences of array A: A000012, A000225(n+1), A001047(n+1), A005061(n+1), A005060(n+1), A005062(n+1), A016169(n+1), A016177(n+1), A016185(n+1), A016189(n+1), A016195(n+1), A016197(n+1).

egf := exp(exp(x)*y + x)*(exp(x)*y - y + 1): ser := series(egf, x, 12):
cx := n -> series(n!*coeff(ser, x, n), y, 12):
Arow := n -> seq(k!*coeff(cx(n), y, k), k=0..9):
for n from 0 to 5 do Arow(n) od; # Peter Luschny, May 10 2021

A[n_, k_] := (k + 1)^(n + 1) - k^(n + 1); Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 10 2021 *)

Array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0.
A(n-1, k-1) = Sum_{j=1} binomial(n, j)*(k-1)^(n-j) = Sum_{j=0} binomial(n, j)*(k-1)^(n-j) - (k-1)^n = (1+(k-1))^n - (k-1)^n = k^n - (k-1)^n (from the combinatorial comment on A(n-1, k-1) above).
O.g.f. row n of array A: RA(n, x) = P(n, x)/(1 - x)^n, with P(n, x) = Sum_{m=0..n} A008292(n+1, m+1)*x^m, (the Eulerian number triangle  A008292 has offset 1) for n >= 0. (See the Oct 26 2008 comment in A047969 by Peter Bala). RA(n, x) = polylog(-(n+1), x)*(1-x)/x. (For P(n, x) see the formula by Vladeta Jovovic, Sep 02 2002.)
E.g.f. of e.g.f.s of the rows of array A: EE(x, y) = exp(x)*(1 + y*(exp(x) - 1))*exp(y*exp(x)), that is A(n, k) = [y^k/k!][x^n/n!] EE(x, y).
Triangle T(n, m) = A(n-m, m) = (m+1)^(n-m+1) - m^(n-m+1), n >= 0, m = 0, 1, ..., n.
E.g.f.: -(exp(x)-1)/(x*exp(x)*y-x). - Vladimir Kruchinin, Nov 02 2022

A020782 Expansion of g.f. 1/((1-7x)(1-8x)(1-9*x)).

Original entry on oeis.org

1, 24, 385, 5160, 62401, 706104, 7628545, 79669320, 810888001, 8089258584, 79415935105, 769621605480, 7379461252801, 70134974713464, 661651583000065, 6203106293141640, 57847125937972801, 537010118353326744, 4965807358070423425, 45765395460943045800, 420553385321258904001

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (24,-191,504).

Cf. A005062, A016149.

CoefficientList[Series[1/((1-7x)(1-8x)(1-9x)),{x,0,20}],x] (* or *) LinearRecurrence[{24,-191,504},{1,24,385},20] (* Harvey P. Dale, Aug 20 2013 *)

If we define f(m,j,x) = Sum_{k=j..m} (binomial(m,k)*stirling2(k,j)*x^(m-k)) then a(n-2) = f(n,2,7), (n>=2). - Milan Janjic, Apr 26 2009
From Vincenzo Librandi, Mar 15 2011: (Start)
a(n) = 24*a(n-1) - 191*a(n-2) + 504*a(n-3), n>=3.
a(n) = 17*a(n-1) - 72*a(n-2) + 7^n,  n>=2. (End)
a(n) = 7^(n+2)/2 - 8^(n+2) + 9^(n+2)/2. - R. J. Mathar, Mar 15 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(7*x)*(49 - 128*exp(x) + 81*exp(2*x))/2.
a(n) = A005062(n+2) - A016149(n+1). (End)

More terms from Elmo R. Oliveira, Mar 26 2025

A248340 a(n) = 10^n - 5^n.

Original entry on oeis.org

0, 5, 75, 875, 9375, 96875, 984375, 9921875, 99609375, 998046875, 9990234375, 99951171875, 999755859375, 9998779296875, 99993896484375, 999969482421875, 9999847412109375, 99999237060546875, 999996185302734375, 9999980926513671875

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (15,-50).

Cf. sequences of the form k^n-5^n: A005062 (k=6), A121213 (k=7), A191468 (k=8), A191466 (k=9), this sequence (k=10), A139743 (k=11).

Cf. A000225, A000351, A011557, A016164.

Magma
```
[10^n-5^n: n in [0..30]];
```

Table[10^n - 5^n, {n,0,30}]
CoefficientList[Series[5 x/((1-5 x)(1-10 x)), {x, 0, 30}], x]

def A248340(n): return pow(10,n) - pow(5,n)
print([A248340(n) for n in range(41)]) # G. C. Greubel, Nov 13 2024

G.f.: 5*x/((1-5*x)*(1-10*x)).
a(n) = 15*a(n-1) - 50*a(n-2).
a(n) = 5^n*(2^n-1) = A000351(n) * A000225(n) = A011557(n) - A000351(n).
a(n) = 5*A016164(n-1).
a(n) = A016164(n) - A011557(n).
E.g.f.: exp(10*x) - exp(5*x). - G. C. Greubel, Nov 13 2024

cp's OEIS Frontend

A016197 a(n) = 12^n - 11^n.

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A327316 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = ((x+r)^n - (x+s)^n)/(r - s), where r = 3 and s = 2.

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A343237 Triangle T obtained from the array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0, by reading antidiagonals upwards.

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A020782 Expansion of g.f. 1/((1-7*x)*(1-8*x)*(1-9*x)).

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A248340 a(n) = 10^n - 5^n.

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Magma

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A020782 Expansion of g.f. 1/((1-7x)(1-8x)(1-9*x)).