A038846 -id:A038846 - cp's OEIS frontend

A081143 5th binomial transform of (0,0,0,1,0,0,0,0,......).

0, 0, 0, 1, 20, 250, 2500, 21875, 175000, 1312500, 9375000, 64453125, 429687500, 2792968750, 17773437500, 111083984375, 683593750000, 4150390625000, 24902343750000, 147857666015625, 869750976562500, 5073547363281250

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, four-fold convolution of A000351 (powers of 5).

With a different offset, number of n-permutations (n=4)of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly three u's. Example: a(4)=20 because we have uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu, xuuu, uuuy, uuyu, uyuu and yuuu. - Zerinvary Lajos, Jun 03 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A038846, A036216, A001789, A081144, A081135.

[5^(n-3) * Binomial(n, 3): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

seq(binomial(n,3)*5^(n-3), n=0..25); # Zerinvary Lajos, Jun 03 2008

CoefficientList[Series[x^3/(1-5x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{20,-150,500,-625}, {0,0,0,1}, 30] (* Harvey P. Dale, Dec 24 2015 *)

vector(31, n, my(m=n-1); 5^(m-3)*binomial(m,3)) \\ G. C. Greubel, Mar 05 2020

[lucas_number2(n, 5, 0)*binomial(n,3)/5^3 for n in range(0, 22)] # Zerinvary Lajos, Mar 12 2009

a(n) = 20*a(n-1) - 150*a(n-2) + 500*a(n-3) - 625*a(n-4), with a(0)=a(1)=a(2)=0, a(3)=1.
a(n) = 5^(n-3)*binomial(n,3).
G.f.: x^3/(1-5*x)^4.
E.g.f.: x^3*exp(x)/6. - G. C. Greubel, Mar 05 2020
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=3} 1/a(n) = 240*log(5/4) - 105/2.
Sum_{n>=3} (-1)^(n+1)/a(n) = 540*log(6/4) - 195/2. (End)

A128962 a(n) = (n^3 - n)*4^n.

Original entry on oeis.org

0, 96, 1536, 15360, 122880, 860160, 5505024, 33030144, 188743680, 1038090240, 5536481280, 28789702656, 146565758976, 732828794880, 3607772528640, 17523466567680, 84112639524864, 399535037743104, 1880164883496960, 8774102789652480, 40637949762600960

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 (16,-96,256,-256).

Cf. A000302, A007531, A036289, A128796.

Cf. A128960, A128961, A128963, A128964, A128965, A128967, A128969.

[(n^3-n)*4^n: n in [1..20]]; // Vincenzo Librandi, Feb 09 2013

CoefficientList[Series[96 x / (1-4 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 09 2013 *)
Table[(n^3-n)4^n,{n,20}] (* or *) LinearRecurrence[{16,-96,256,-256},{0,96,1536,15360},20] (* Harvey P. Dale, Dec 31 2018 *)

G.f.: 96*x^2/(1-4*x)^4. - Vincenzo Librandi, Feb 09 2013
a(n) = 16*a(n-1) - 96*a(n-2) + 256*a(n-3) - 256*a(n-4). - Vincenzo Librandi, Feb 09 2013
a(n) = 96*A038846(n-2) for n>1. - Bruno Berselli, Feb 10 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000302(n).
Sum_{n>=2} 1/a(n) = (9/8)*log(4/3) - 5/16.
Sum_{n>=2} (-1)^n/a(n) = (25/8)*log(5/4) - 11/16. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

A006044 a(n) = 4^(n-4)(n-1)(n-2)*(n-3).

Original entry on oeis.org

6, 96, 960, 7680, 53760, 344064, 2064384, 11796480, 64880640, 346030080, 1799356416, 9160359936, 45801799680, 225485783040, 1095216660480, 5257039970304, 24970939858944, 117510305218560, 548381424353280, 2539871860162560, 11683410556747776, 53409876830846976

Offset: 4

N. J. A. Sloane

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 = 4..1000
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 (16,-96,256,-256).

Cf. A000142, A006043, A038846, A154120.

Column k=3 of square array A152818. - Paul Curtz, Dec 17 2008 [corrected by Omar E. Pol, Jan 07 2009]

[4^(n-4)*(n-3)*(n-2)*(n-1): n in [4..30]]; // Vincenzo Librandi, Aug 14 2011

a[n_] := 4^(n - 4)*(n - 1)*(n - 2)*(n - 3); Array[a, 25, 4] (* Amiram Eldar, Jan 08 2023 *)

G.f. = 6*x^4/(1-4*x)^4. - Emeric Deutsch, Apr 29 2004
a(n) = 6*A038846(n). - R. J. Mathar , Mar 22 2013
E.g.f.: (3 + exp(4*x)*(32*x^3 - 24*x^2 + 12*x - 3))/128. - Stefano Spezia, Jan 01 2023
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=4} 1/a(n) = 18*log(4/3) - 5.
Sum_{n>=4} (-1)^n/a(n) = 50*log(5/4) - 11. (End)

More terms from Emeric Deutsch, Apr 29 2004
Erroneous reference deleted by Martin J. Erickson (erickson(AT)truman.edu), Nov 03 2010
Entry revised by N. J. A. Sloane, Dec 27 2021

A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

Original entry on oeis.org

1, 24, 336, 3584, 32256, 258048, 1892352, 12976128, 84344832, 524812288, 3148873728, 18320719872, 103817412608, 574988746752, 3121367482368, 16647293239296, 87398289506304, 452414675091456, 2312341672689664, 11683410556747776, 58417052783738880, 289303499500421120

Offset: 0

Wolfdieter Lang

Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).
With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - Zerinvary Lajos, Jun 16 2008
Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-240,1280,-3840,6144,-4096).

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+5,5)); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+5,5)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* Harvey P. Dale, Mar 24 2018 *)

Vec(1/(1-4*x)^6 + O(x^30)) \\ Michel Marcus, Aug 21 2015

[lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+5, 5)*4^n.
G.f.: 1/(1-4*x)^6.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.
Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)

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.

A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 9, 1, 1, 32, 54, 16, 1, 1, 80, 270, 160, 25, 1, 1, 192, 1215, 1280, 375, 36, 1, 1, 448, 5103, 8960, 4375, 756, 49, 1, 1, 1024, 20412, 57344, 43750, 12096, 1372, 64, 1

Offset: 0

Paul Curtz, Jan 08 2009

From A152650/A152656,coefficients of other exponential polynomials(*). a(n) is triangle A152818 where terms of each column is divided by the beginning one. See A000004, A001787(n+1), A006043=2*A027472, A006044=6*A038846.
(*) Not factorial as written in A006044. See A000110, Bell-Touchard. Second diagonal is 1,4,9,16,25, denominators of Lyman's spectrum of hydrogen, A000290(n+1) which has homogeneous indices for denominators series of Rydberg-Ritz spectrum of hydrogen.
The matrix inverse starts
      1;
     -1,      1;
      3,     -4,     1;
    -16,     24,    -9,     1;
    125,   -200,    90,   -16,   1;
  -1296,   2160, -1080,   240, -25,   1;
  16807, -28812, 15435, -3920, 525, -36, 1;
.. compare with A122525 (row reversed). - R. J. Mathar, Mar 22 2013
From Peter Bala, Jan 14 2015: (Start)
Exponential Riordan array [exp(z), z*exp(z)]. This triangle is the particular case a = 0, b = 1, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. Cf. A059297.
This is the triangle of connection constants when expressing the monomials x^n as a linear combination of the basis polynomials (x - 1)*(x - k - 1)^(k-1), k = 0,1,2,.... For example, from row 3 we have x^3 = 1 + 12*(x - 1) + 9*(x - 1)*(x - 3) + (x - 1)*(x - 4)^2.
Let M be the infinite lower unit triangular array with (n,k)-th entry (k*(n - k + 1) + 1)/(k + 1)*binomial(n,k). M is the row reverse of A145033. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle. See the Example section. (End)
T(n,k) is also the number of idempotent partial transformations of {1,2,...,n} having exactly k fixed points. - Geoffrey Critzer, Nov 25 2021

			With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1        \      /1        \
|1 1     ||0 1       ||0 1      |      |1  1      |
|1 3 1   ||0 1 1     ||0 0 1    |... = |1  4  1   |
|1 6 5 1 ||0 1 3 1   ||0 0 1 1  |      |1 12  9  1|
|...     ||0 1 6 5 1 ||0 0 1 3 1|      |...       |
|...     ||...       ||...      |      |          |
- _Peter Bala_, Jan 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1274
Peter Bala, A 3 parameter family of generalized Stirling numbers
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A080108, A152818, A059297, A145033.

/* As triangle */ [[(k+1)^(n-k)*Binomial(n,k) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 15 2016

T[n_, k_] := (k + 1)^(n - k)*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 15 2016 *)

T(n,k) = (k+1)^(n-k)*binomial(n,k). k!*T(n,k) gives the entries for A152818 read as a triangular array.
E.g.f.: exp(x*(1+t*exp(x))) = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + (1+12*t+9*t^2+t*3)*x^3/3! + .... O.g.f.: Sum_{k>=1} (t*x)^(k-1)/(1-k*x)^k = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... Row sums are A080108. - Peter Bala, Oct 09 2011
From Peter Bala, Jan 14 2015: (Start)
Recurrence equation: T(n+1,k+1) = T(n,k+1) + Sum_{j = 0..n-k} (j + 1)*binomial(n,j)*T(n-j,k) with T(n,0) = 1 for all n.
Equals the matrix product A007318 * A059297. (End)

A129004 a(n) = (n^3 + n^2)*4^n.

Original entry on oeis.org

8, 192, 2304, 20480, 153600, 1032192, 6422528, 37748736, 212336640, 1153433600, 6090129408, 31406948352, 158779572224, 789200240640, 3865470566400, 18691697672192, 89369679495168, 423037098786816, 1984618488135680, 9235897673318400, 42669847250731008, 195836215046438912

Offset: 1

Mohammad K. Azarian, May 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A036289, A038846, A128796.

[(n^3+n^2)*4^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

Table[(n^3+n^2)4^n,{n, 20}] (* or *) LinearRecurrence[{16,-96,256,-256}, {8,192,2304,20480},20] (* Harvey P. Dale, May 12 2011 *)

G.f.: 8*x*(1+8*x)/(1-4*x)^4. - R. J. Mathar, Dec 19 2008
a(1)=8, a(2)=192, a(3)=2304, a(4)=20480, a(n)=16*a(n-1)-96*a(n-2)+ 256*a(n-3)-256*a(n-4). - Harvey P. Dale, May 12 2011
a(n) = 8*(A038846(n-1)+8*A038846(n-2)), with A038846(-1)=0. - Bruno Berselli, Feb 12 2013
E.g.f.: 8*exp(4*x)*x*(1 + 8*x + 8*x^2). - Stefano Spezia, Sep 02 2024

A172978 a(n) = binomial(n+10, 10)*4^n.

Original entry on oeis.org

1, 44, 1056, 18304, 256256, 3075072, 32800768, 318636032, 2867724288, 24216338432, 193730707456, 1479398129664, 10848919617536, 76776969601024, 526470648692736, 3509804324618240, 22813728110018560, 144934272698941440, 901813252348968960, 5505807224867389440

Offset: 0

Zerinvary Lajos, Feb 06 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..157
Index entries for linear recurrences with constant coefficients, signature (44,-880,10560,-84480,473088,-1892352,5406720,-10813440,14417920,-11534336,4194304).

Cf. A002697, A038845, A038846, A040075, A045543, A054337, A054338, A054339, A054340

[Binomial(n+10, 10)*4^n: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

Table[Binomial[n + 10, 10]*4^n, {n, 0, 20}]

From Amiram Eldar, Mar 27 2022: (Start)
G.f.: 1/(1 - 4*x)^11.
Sum_{n>=0} 1/a(n) = 14269429/63 - 787320*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 78125000*log(5/4) - 1098284605/63. (End)

A293270 a(n) = n^nbinomial(2n-1, n).

Original entry on oeis.org

1, 1, 12, 270, 8960, 393750, 21555072, 1413199788, 107961384960, 9418192087590, 923780000000000, 100633991211229476, 12055263261877075968, 1575041416811693275900, 222887966509090352332800, 33962507149515380859375000, 5543988061027763016035205120

Offset: 0

Ilya Gutkovskiy, Oct 04 2017

nonn

The n-th term of the n-fold convolution of the powers of n.

Cf. A000312, A001700, A088218.

Cf. A001787, A027472, A036071, A036084, A036226, A038846, A053107, A053108, A053109.

Join[{1}, Table[n^n Binomial[2 n - 1, n], {n, 1, 16}]]
Join[{1}, Table[(-1)^n n^n Binomial[-n, n], {n, 1, 16}]]
Table[SeriesCoefficient[1/(1 - n x)^n, {x, 0, n}], {n, 0, 16}]

a(n) = n^n*binomial(2*n-1, n); \\ Altug Alkan, Oct 04 2017

a(n) = [x^n] 1/(1 - n*x)^n.

a(n) ~ 2^(2*n-1)*n^n/sqrt(Pi*n).

A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*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, 4, 16, 1, 64, 8, 256, 48, 1, 1024, 256, 12, 4096, 1280, 96, 1, 16384, 6144, 640, 16, 65536, 28672, 3840, 160, 1, 262144, 131072, 21504, 1280, 20, 1048576, 589824, 114688, 8960, 240, 1, 4194304, 2621440, 589824, 57344, 2240, 24, 16777216, 11534336, 2949120, 344064, 17920, 336, 1

Offset: 0

Shara Lalo, Jun 11 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013611 ((1+4*x)^n).
The coefficients in the expansion of 1/(1-4x-x^2) are given by the sequence generated by the row sums.
The row sums are A001076 (Denominators of continued fraction convergent to sqrt(5)).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 4.236067977...; a metallic mean (see A098317), when n approaches infinity.

			Triangle begins:
         1;
         4;
        16,        1;
        64,        8;
       256,       48,        1;
      1024,      256,       12;
      4096,     1280,       96,       1;
     16384,     6144,      640,      16;
     65536,    28672,     3840,     160,      1;
    262144,   131072,    21504,    1280,     20;
   1048576,   589824,   114688,    8960,    240,    1;
   4194304,  2621440,   589824,   57344,   2240,   24;
  16777216, 11534336,  2949120,  344064,  17920,  336,  1;
  67108864, 50331648, 14417920, 1966080, 129024, 3584, 28;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 90, 373.

Shara Lalo, Left justified triangle
Shara Lalo, Skew diagonals in triangle A013611

Row sums give A001076.
Cf. A000302 (column 0), A002697 (column 1), A038845 (column 2), A038846 (column 3), A040075 (column 4).
Cf. A013611.
Cf. A098317.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 4 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

G.f.: 1 / (1 - 4*t*x - t^2).

cp's OEIS Frontend

A081143 5th binomial transform of (0,0,0,1,0,0,0,0,......).

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A128962 a(n) = (n^3 - n)*4^n.

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A006044 a(n) = 4^(n-4)*(n-1)*(n-2)*(n-3).

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A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

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

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A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

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A129004 a(n) = (n^3 + n^2)*4^n.

A006044 a(n) = 4^(n-4)(n-1)(n-2)*(n-3).

A293270 a(n) = n^nbinomial(2n-1, n).