A027472 -id:A027472 - cp's OEIS frontend

A036223 Expansion of 1/(1-3*x)^10; 10-fold convolution of A000244 (powers of 3).

1, 30, 495, 5940, 57915, 486486, 3648645, 25019280, 159497910, 956987460, 5454828522, 29753610120, 156206453130, 793048146660, 3908594437110, 18761253298128, 87943374834975, 403504896301650, 1815772033357425

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n >= 9) of 4 objects: u, v, z, x with repetition allowed, containing exactly nine (9) u's. - Zerinvary Lajos, Jul 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049).

Cf. A027465.

Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), this sequence (m=9), A172362 (m=10).

[3^n*Binomial(n+9, 9): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+9, 9), n=0..20); # Zerinvary Lajos, Jul 02 2008

Table[3^n*Binomial[n+9,9], {n,0,30}] (* G. C. Greubel, May 18 2021 *)
CoefficientList[Series[1/(1-3x)^10,{x,0,30}],x] (* or *) LinearRecurrence[ {30,-405,3240,-17010,61236,-153090,262440,-295245,196830,-59049},{1,30,495,5940,57915,486486,3648645,25019280,159497910,956987460},30] (* Harvey P. Dale, Jan 16 2022 *)

[3^n*binomial(n+9,9) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

a(n) = 3^n*binomial(n+9, 9).
a(n) = A027465(n+10, 10).
G.f.: 1/(1-3*x)^10.
E.g.f.: (4480 + 120960*x + 725760*x^2 + 1693440*x^3 + 1905120*x^4 + 1143072*x^5 + 381024*x^6 + 69984*x^7 + 6561*x^8 + 243*x^9)*exp(3*x)/4480. - G. C. Greubel, May 18 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 6912*log(3/2) - 784431/280.
Sum_{n>=0} (-1)^n/a(n) = 1769472*log(4/3) - 142532433/280. (End)

A038221 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)3^j.

Original entry on oeis.org

1, 3, 3, 9, 18, 9, 27, 81, 81, 27, 81, 324, 486, 324, 81, 243, 1215, 2430, 2430, 1215, 243, 729, 4374, 10935, 14580, 10935, 4374, 729, 2187, 15309, 45927, 76545, 76545, 45927, 15309, 2187, 6561, 52488, 183708, 367416, 459270, 367416, 183708, 52488, 6561

Offset: 0

N. J. A. Sloane

Triangle of coefficients in expansion of (3 + 3x)^n = 3^n (1 +x)^n, where n is a nonnegative integer. (Coefficients in expansion of (1 +x)^n are given in A007318: Pascal's triangle). - Zagros Lalo, Jul 23 2018

			Triangle begins as:
     1;
     3,     3;
     9,    18,      9;
    27,    81,     81,     27;
    81,   324,    486,    324,     81;
   243,  1215,   2430,   2430,   1215,    243;
   729,  4374,  10935,  14580,  10935,   4374,    729;
  2187, 15309,  45927,  76545,  76545,  45927,  15309,  2187;
  6561, 52488, 183708, 367416, 459270, 367416, 183708, 52488, 6561;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Indranil Ghosh, Rows 0..100 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A007318, A304236, A304249.
Cf. A000007, A000400, A027465, A030195, A057083.
Columns k: A000244 (k=0), 3*A027471 (k=1), 3^2*A027472 (k=2), 3^3*A036216 (k=3), 3^4*A036217 (k=4), 3^5*A036219 (k=5), 3^6*A036220 (k=6), 3^7*A036221 (k=7), 3^8*A036222 (k=8), 3^9*A036223 (k=9), 3^10*A172362 (k=10).

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*3^(i-j)*3^j))); # Muniru A Asiru, Jul 23 2018

a038221 n = a038221_list !! n
a038221_list = concat $ iterate ([3,3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

[3^n*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 17 2022

(* programs from Zagros Lalo, Jul 23 2018 *)
t[0, 0]=1; t[n_, k_]:= t[n, k]= If[n<0 || k<0, 0, 3 t[n-1, k] + 3 t[n-1, k-1]]; Table[t[n, k], {n,0,10}, {k,0,n}]//Flatten
Table[CoefficientList[Expand[3^n *(1+x)^n], x], {n,0,10}]//Flatten
Table[3^n Binomial[n, k], {n,0,10}, {k,0,n}]//Flatten  (* End *)

def A038221(n,k): return 3^n*binomial(n,k)
flatten([[A038221(n,k) for k in range(n+1)] for n in range(10)]) # G. C. Greubel, Oct 17 2022

G.f.: 1/(1 - 3*x - 3*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 3 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018
From G. C. Greubel, Oct 17 2022: (Start)
T(n, k) = T(n, n-k).
T(n, n) = A000244(n).
T(n, n-1) = 3*A027471(n).
T(n, n-2) = 9*A027472(n+1).
T(n, n-3) = 27*A036216(n-3).
T(n, n-4) = 81*A036217(n-4).
T(n, n-5) = 243*A036219(n-5).
Sum_{k=0..n} T(n, k) = A000400(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A030195(n+1), n >= 0.
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A057083(n).
T(n, k) = 3^k * A027465(n, k). (End)

A172362 a(n) = binomial(n+10, 10)*3^n.

Original entry on oeis.org

1, 33, 594, 7722, 81081, 729729, 5837832, 42532776, 287096238, 1818276174, 10909657044, 62482581252, 343654196886, 1824010737318, 9380626649064, 46903133245320, 228652774570935, 1089463220014455, 5084161693400790

Offset: 0

Zerinvary Lajos, Feb 01 2010

With a different offset, number of n-permutations (n>=10) of 4 objects: u, v, z, x with repetition allowed, containing exactly ten, (10) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (33,-495,4455,-26730,112266,-336798,721710,-1082565,1082565,-649539,177147).

Cf. A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223.

[3^n*Binomial(n+10, 10): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple
```
seq(binomial(n+10, 10)*3^n, n=0..30);
```

Table[Binomial[n + 10, 10]*3^n, {n, 0, 20}]

G.f.: 1/(1-3*x)^11. - Vincenzo Librandi, Oct 15 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 261617/42 - 15360*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 7864320*log(4/3) - 47510881/21. (End)

A081130 Square array of binomial transforms of (0,0,1,0,0,0,...), read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 6, 6, 0, 0, 0, 1, 9, 24, 10, 0, 0, 0, 1, 12, 54, 80, 15, 0, 0, 0, 1, 15, 96, 270, 240, 21, 0, 0, 0, 1, 18, 150, 640, 1215, 672, 28, 0, 0, 0, 1, 21, 216, 1250, 3840, 5103, 1792, 36, 0, 0, 0, 1, 24, 294, 2160, 9375, 21504, 20412, 4608, 45, 0

Offset: 0

Paul Barry, Mar 08 2003

Rows, of the square array, are three-fold convolutions of sequences of powers.

			The array begins as:
  0,  0,  0,   0,   0,    0, ...
  0,  0,  0,   0,   0,    0, ...
  0,  1,  1,   1,   1,    1, ... A000012
  0,  3,  6,   9,  12,   15, ... A008585
  0,  6, 24,  54,  96,  150, ... A033581
  0, 10, 80, 270, 640, 1250, ... A244729
The antidiagonal triangle begins as:
  0;
  0, 0;
  0, 0, 0;
  0, 0, 1, 0;
  0, 0, 1, 3,  0;
  0, 0, 1, 6,  6,  0;
  0, 0, 1, 9, 24, 10, 0;

G. C. Greubel, Antidiadoganal rows n = 0..50, flattened

Main diagonal: A081131.
Rows: A000012 (n=2), A008585 (n=3), A033581 (n=4), A244729 (n=5).
Columns: A000217 (k=1), A001788 (k=2), A027472 (k=3), A038845 (k=4), A081135 (k=5), A081136 (k=6), A027474 (k=7), A081138 (k=8), A081139 (k=9), A081140 (k=10), A081141 (k=11), A081142 (k=12), A027476 (k=15).

[k eq n select 0 else (n-k)^(k-2)*Binomial(k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021

Table[If[k==n, 0, (n-k)^(k-2)*Binomial[k, 2]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 14 2021 *)

T(n, k)=if (k==0, 0, k^(n-2)*binomial(n, 2));
seq(nn) = for (n=0, nn, for (k=0, n, print1(T(k, n-k), ", ")); );
seq(12) \\ Michel Marcus, May 14 2021

flatten([[0 if (k==n) else (n-k)^(k-2)*binomial(k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021

T(n, k) = k^(n-2)*binomial(n, 2), with T(n, 0) = 0 (square array).
T(n, n) = A081131(n).
Rows have g.f. x^3/(1-k*x)^n.
From G. C. Greubel, May 14 2021: (Start)
T(k, n-k) = (n-k)^(k-2)*binomial(k,2) with T(n, n) = 0 (antidiagonal triangle).
Sum_{k=0..n} T(n, n-k) = A081197(n). (End)

Term a(5) corrected by G. C. Greubel, May 14 2021

A006043 A traffic light problem: expansion of 2/(1 - 3*x)^3.

Original entry on oeis.org

2, 18, 108, 540, 2430, 10206, 40824, 157464, 590490, 2165130, 7794468, 27634932, 96722262, 334807830, 1147912560, 3902902704, 13172296626, 44165935746, 147219785820, 488149816140, 1610894393262, 5292938720718, 17322344904168, 56485907296200, 183579198712650, 594796603828986

Offset: 0

N. J. A. Sloane, Simon Plouffe

Column 2 of square array A152818. - Omar E. Pol, Jan 05 2009

In [Bach et al., Section 9], 2*a(n-2) counts the "small diagrams". - Eric M. Schmidt, Sep 23 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Bach, Jeremie Dusart, Lisa Hellerstein, and Devorah Kletenik, Submodular Goal Value of Boolean Functions, arXiv:1702.04067 [cs.DM], 2017.
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
Frank A. Haight, Letter to N. J. A. Sloane, n.d..
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A006044, A000142, A152818, A154120. - Omar E. Pol, Jan 05 2009

Cf. A000217, A000244, A027472, A116138,

[(n+2)*(n+1)*3^n: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

seq((n+2)*(n+1)*3^n, n=0..23); # Zerinvary Lajos, Apr 25 2007

f[n_] := (n + 2) (n + 1) 3^n; Array[f, 22, 0] (* Robert G. Wilson v, Mar 15 2011 *)
CoefficientList[Series[2/(1 - 3 x)^3, {x, 0, 21}], x] (* Robert G. Wilson v, Mar 15 2011 *)
LinearRecurrence[{9,-27,27},{2,18,108},30] (* Harvey P. Dale, Apr 27 2017 *)

a(n)=(n+2)*(n+1)*3^n \\ Charles R Greathouse IV, Mar 16 2011

a(n) = (n+2)*(n+1)*3^n. - Zerinvary Lajos, Apr 25 2007, corrected by R. J. Mathar, Mar 14 2011
a(n) = 2*A027472(n+3) = A116138(n+1)/3. - R. J. Mathar, Mar 14 2011
a(n) = 2*A000217(n+1)*A000244(n). - Zak Seidov, Mar 14 2011
E.g.f.: exp(3*x)*(2 + 12*x + 9*x^2). - Stefano Spezia, Jan 01 2023
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 3 - 6*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 12*log(4/3) - 3. (End)

A036068 Expansion of (-1+1/(1-3x)^3)/(9x).

Original entry on oeis.org

1, 6, 30, 135, 567, 2268, 8748, 32805, 120285, 433026, 1535274, 5373459, 18600435, 63772920, 216827928, 731794257, 2453663097, 8178876990, 27119434230, 89494132959, 294052151151, 962352494676, 3138105960900, 10198844372925

Offset: 0

Wolfdieter Lang

G.f. for a(n)=A027472(n+3), n >= 0, is 1/(1-3*x)^3.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Index entries for linear recurrences with constant coefficients, signature (9, -27, 27).

Cf. A001792, A027472. a(n)= A030524(n+1, 1) (first column of triangle).

CoefficientList[Series[((1/(1-3x))^3-1)/(9x),{x,0,30}],x] (* Harvey P. Dale, Nov 26 2018 *)

a(n) = 3^(n-1)*binomial(n+3, 2); G.f.: (-1+(1-3*x)^(-3))/(x*3^2)=(1-3*x+3*x^2)/(1-3*x)^3.
G.f.: F(4,1;2;3x); [From Paul Barry, Sep 03 2008]
D-finite with recurrence: (n+1)*a(n) +3*(-n-3)*a(n-1)=0. - R. J. Mathar, Jan 28 2020

A099097 Riordan array (1, 3+x).

Original entry on oeis.org

1, 0, 3, 0, 1, 9, 0, 0, 6, 27, 0, 0, 1, 27, 81, 0, 0, 0, 9, 108, 243, 0, 0, 0, 1, 54, 405, 729, 0, 0, 0, 0, 12, 270, 1458, 2187, 0, 0, 0, 0, 1, 90, 1215, 5103, 6561, 0, 0, 0, 0, 0, 15, 540, 5103, 17496, 19683, 0, 0, 0, 0, 0, 1, 135, 2835, 20412, 59049, 59049, 0, 0, 0, 0, 0, 0, 18, 945, 13608, 78732, 196830, 177147

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A006190(n+1). Diagonal sums are A052931. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/3, -1/3, 0, 0, 0, 0, 0, ...] DELTA [3, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008

			Triangle begins:
  1;
  0, 3;
  0, 1, 9;
  0, 0, 6, 27;
  0, 0, 1, 27,  81;
  0, 0, 0,  9, 108, 243;
  ...

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

Cf. A027465.

Diagonals are of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

Table[3^(2*k-n)*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 19 2021 *)

flatten([[3^(2*k-n)*binomial(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 19 2021

Triangle: T(n, k) = binomial(k, n-k)*3^k*(1/3)^(n-k).
G.f. of column k: (3*x + x^2)^k.
G.f.: 1/(1 - 3*y*x - y*x^2). - Philippe Deléham, Nov 21 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A006190(n+1), A135030(n+1), A181353(n+1) for x = 0,1,2,3 respectively. - Philippe Deléham, Nov 21 2011

A129530 a(n) = (1/2)n(n-1)*3^(n-1).

Original entry on oeis.org

0, 0, 3, 27, 162, 810, 3645, 15309, 61236, 236196, 885735, 3247695, 11691702, 41452398, 145083393, 502211745, 1721868840, 5854354056, 19758444939, 66248903619, 220829678730, 732224724210, 2416341589893, 7939408081077

Offset: 0

Emeric Deutsch, Apr 22 2007

Number of inversions in all ternary words of length n on {0,1,2}. Example: a(2)=3 because each of the words 10,20,21 has one inversion and the words 00,01,02,11,12,22 have no inversions. a(n)=3*A027472(n+1). a(n)=Sum(k*A129529(n,k),k>=0).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A027472, A129529, A001788, A129532.

Maple
```
seq(n*(n-1)*3^(n-1)/2,n=0..27);
```

Table[(n(n-1)3^(n-1))/2,{n,0,30}] (* or *) LinearRecurrence[{9,-27,27},{0,0,3},30] (* Harvey P. Dale, Dec 18 2013 *)

a(n)=n*(n-1)*3^(n-1)/2 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: 3x^2/(1-3x)^3.
a(0)=0, a(1)=0, a(2)=3, a(n)=9*a(n-1)-27*a(n-2)+27*a(n-3). - Harvey P. Dale, Dec 18 2013
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=2} 1/a(n) = 2 * (1 - 2 * log(3/2)).
Sum_{n>=2} (-1)^n/a(n) = 2*(4*log(4/3) - 1). (End)
a(n) = 3*A027472(n+1). - R. J. Mathar, Jul 26 2022

A317497 Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

Original entry on oeis.org

1, 3, 9, 27, 1, 81, 6, 243, 27, 729, 108, 1, 2187, 405, 9, 6561, 1458, 54, 19683, 5103, 270, 1, 59049, 17496, 1215, 12, 177147, 59049, 5103, 90, 531441, 196830, 20412, 540, 1, 1594323, 649539, 78732, 2835, 15, 4782969, 2125764, 295245, 13608, 135, 14348907, 6908733, 1082565, 61236, 945, 1

Offset: 0

Zagros Lalo, Jul 31 2018

The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n) and  along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)
The coefficients in the expansion of 1/(1-3x-x^3) are given by the sequence generated by the row sums.
The row sums give A052541.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.1038034027355..., when n approaches infinity.

			Triangle begins:
         1;
         3;
         9;
        27,        1;
        81,        6;
       243,       27;
       729,      108,       1;
      2187,      405,       9;
      6561,     1458,      54;
     19683,     5103,     270,      1;
     59049,    17496,    1215,     12;
    177147,    59049,    5103,     90;
    531441,   196830,   20412,    540,    1;
   1594323,   649539,   78732,   2835,   15;
   4782969,  2125764,  295245,  13608,  135;
  14348907,  6908733, 1082565,  61236,  945,  1;
  43046721, 22320522, 3897234, 262440, 5670, 18;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 364-366

G. C. Greubel, Rows n = 0..120 of the irregular triangle, flattened
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3x)^n
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n

Row sums give A052541.
Cf. A013610, A027465, A317496, A304236, A304249.
Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3), A036217 (column 4).
Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), A304249 (q=2), this sequence (q=3), A318773 (q=4).

Flat(List([0..16],n->List([0..Int(n/3)],k->3^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Aug 01 2018

[3^(n-3*k)*Binomial(n-2*k,k): k in [0..Floor(n/3)], n in [0..24]]; // G. C. Greubel, May 12 2021

T[n_, k_]:= T[n, k] = 3^(n-3k)(n-2k)!/((n-3k)! k!); Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]//Flatten
T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, 3 T[n-1, k] + T[n-3, k-1]]; Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}]//Flatten

flatten([[3^(n-3*k)*binomial(n-2*k,k) for k in (0..n//3)] for n in (0..24)]) # G. C. Greubel, May 12 2021

T(n,k) = 3^(n-3*k) * (n-2*k)!/(k! * (n-3*k)!) where n is a nonnegative integer and k = 0..floor(n/3).

A053109 Expansion of 1/(1-10*x)^10.

Original entry on oeis.org

1, 100, 5500, 220000, 7150000, 200200000, 5005000000, 114400000000, 2431000000000, 48620000000000, 923780000000000, 16796000000000000, 293930000000000000, 4974200000000000000, 81719000000000000000

Offset: 0

Wolfdieter Lang

This is the tenth member of the k-family of sequences a(k,n) := k^n*binomial(n+k-1,k-1) starting with A000012 (powers of 1), A001787(n+1), A027472(n+3), A038846, A036071, A036084, A036226, A053107-9 for k=1..10.

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (100, -4500, 120000, -2100000, 25200000, -210000000, 1200000000, -4500000000, 10000000000, -10000000000).

List([0..15],n->10^n*Binomial(n+9,9)); # Muniru A Asiru, Aug 16 2018

[10^n*Binomial(n+9, 9): n in [0..30]]; // G. C. Greubel, Aug 16 2018

seq(coeff(series(1/(1-10*x)^10, x, n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Aug 16 2018

CoefficientList[Series[1/(1-10x)^10,{x,0,20}],x] (* or *) Table[10^n Binomial[n+9,9],{n,0,20}] (* Harvey P. Dale, May 19 2011 *)

vector(30,n,n--; 10^n*binomial(n+9, 9)) \\ G. C. Greubel, Aug 16 2018

[lucas_number2(n, 10, 0)*binomial(n,9)/10 ^9 for n in range(9, 24)] # Zerinvary Lajos, Mar 13 2009

a(n) = 10^n*binomial(n+9, 9);

G.f.: 1/(1-10*x)^10.

cp's OEIS Frontend

A036223 Expansion of 1/(1-3*x)^10; 10-fold convolution of A000244 (powers of 3).

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A038221 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.

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A172362 a(n) = binomial(n+10, 10)*3^n.

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A081130 Square array of binomial transforms of (0,0,1,0,0,0,...), read by antidiagonals.

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A006043 A traffic light problem: expansion of 2/(1 - 3*x)^3.

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A036068 Expansion of (-1+1/(1-3*x)^3)/(9*x).

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A038221 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)3^j.

A036068 Expansion of (-1+1/(1-3x)^3)/(9x).

A129530 a(n) = (1/2)n(n-1)*3^(n-1).