A081144 -id:A081144 - cp's OEIS frontend

A128964 a(n) = (n^3-n)*6^n.

0, 216, 5184, 77760, 933120, 9797760, 94058496, 846526464, 7255941120, 59861514240, 478892113920, 3735358488576, 28524555730944, 213934167982080, 1579821548175360, 11510128422420480, 82872924641427456, 590469588070170624, 4168020621671792640, 29176144351702548480

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 (24,-216,864,-1296).

Cf. A000400, A007531, A036289, A081144, A128796.

Cf. A128960, A128961, A128962, A128963, A128965, A128967, A128969.

[(n^3-n)*6^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

I:=[0, 216, 5184, 77760]; [n le 4 select I[n] else 24*Self(n-1) -216*Self(n-2) +864*Self(n-3) -1296*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

CoefficientList[Series[216 x/(1 - 6 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 216*x^2/(1-6*x)^4.
a(n) = 216*A081144(n+1). (End)
a(n) = 24*a(n-1) - 216*a(n-2) + 864*a(n-3) - 1296*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=2} 1/a(n) = 25*log(6/5)/12 - 3/8.
Sum_{n>=2} (-1)^n/a(n) = 49*log(7/6)/12 - 5/8. (End)
a(n) = A007531(n+1)*A000400(n). - Amiram Eldar, Oct 02 2022

Corrected offset. - Mohammad K. Azarian, Nov 20 2008

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

Original entry on oeis.org

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)

A172501 a(n) = binomial(n+8,8)*6^n.

Original entry on oeis.org

1, 54, 1620, 35640, 641520, 10007712, 140107968, 1801388160, 21616657920, 244988789760, 2645878929408, 27420927086592, 274209270865920, 2657720625315840, 25058508752977920, 230538280527396864, 2074844524746571776, 18307451688940339200, 158664581304149606400

Offset: 0

Zerinvary Lajos, Feb 05 2010

With a different offset, number of n-permutations (n>=8) of 7 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..400
Index entries for linear recurrences with constant coefficients, signature (54,-1296,18144,-163296,979776,-3919104,10077696,-15116544,10077696).

Cf. A081136, A081144, A139626, A036084, A050988, A141407.

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

Table[Binomial[n + 8, 8]*6^n, {n, 0, 20}]

Vec(1 / (1 - 6*x)^9 + O(x^30)) \\ Colin Barker, Jul 24 2017

From Colin Barker, Jul 24 2017: (Start)
G.f.: 1 / (1 - 6*x)^9.
a(n) = (2^(-7 + n)*3^(-2 + n)*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n)) / 35.
(End)
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 4785948/7 - 3750000*log(6/5).
Sum_{n>=0} (-1)^n/a(n) = 39530064*log(7/6) - 213275484/35. (End)

A173123 a(n) = binomial(n+9,9)*6^n.

Original entry on oeis.org

1, 60, 1980, 47520, 926640, 15567552, 233513280, 3202467840, 40831464960, 489977579520, 5585744406528, 60935393525760, 639821632020480, 6496650417438720, 64038411257610240, 614768748073058304, 5763457013184921600, 52888193768049868800, 475993743912448819200

Offset: 0

Zerinvary Lajos, Feb 10 2010

With a different offset, number of n-permutations (n>=9) of 7 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly 9 u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (60,-1620,25920,-272160,1959552,-9797760,33592320,-75582720,100776960,-60466176).

Cf. A081136, A081144, A139626, A036084, A050988, A141407, A172501.

[6^n* Binomial(n+9, 9): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 9, 9]*6^n, {n, 0, 20}]

a(n) = C(n + 9, 9)*6^n.
From Chai Wah Wu, Nov 12 2021: (Start)
a(n) = 60*a(n-1) - 1620*a(n-2) + 25920*a(n-3) - 272160*a(n-4) + 1959552*a(n-5) - 9797760*a(n-6) + 33592320*a(n-7) - 75582720*a(n-8) + 100776960*a(n-9) - 60466176*a(n-10) for n > 9.
G.f.: 1/(6*x - 1)^10. (End)
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 21093750*log(6/5) - 107683641/28.
Sum_{n>=0} (-1)^n/a(n) = 311299254*log(7/6) - 959739813/20. (End)

A173124 a(n) = binomial(n+10,10)*6^n.

Original entry on oeis.org

1, 66, 2376, 61776, 1297296, 23351328, 373621248, 5444195328, 73496636928, 930957401088, 11171488813056, 127964326404096, 1407607590445056, 14942295960109056, 153692187018264576, 1536921870182645760, 14984988234280796160, 142798123173734645760, 1332782482954856693760

Offset: 0

Zerinvary Lajos, Feb 10 2010

With a different offset, number of n-permutations (n>=10) of 7 objects: r, s, t, u, v, z, x, with repetition allowed, containing exactly 10 u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (66,-1980,35640,-427680,3592512,-21555072,92378880,-277136640,554273280,-665127936,362797056).

Cf. A081136, A081144, A139626, A036084, A050988, A141407, A172501, A173123.

[6^n* Binomial(n+10, 10): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 10, 10]*6^n, {n, 0, 20}]

From Chai Wah Wu, Nov 12 2021: (Start)
a(n) = 66*a(n-1) - 1980*a(n-2) + 35640*a(n-3) - 427680*a(n-4) + 3592512*a(n-5) - 21555072*a(n-6) + 92378880*a(n-7) - 277136640*a(n-8) + 554273280*a(n-9) - 665127936*a(n-10) + 362797056*a(n-11) for n > 10.
G.f.: -1/(6*x - 1)^11. (End)
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=0} 1/a(n) = 897363955/42 - 117187500*log(6/5).
Sum_{n>=0} (-1)^n/a(n) = 2421216420*log(7/6) - 2239392937/6. (End)

A304255 Triangle read by rows: T(0,0) = 1; T(n,k) = 6*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, 6, 36, 1, 216, 12, 1296, 108, 1, 7776, 864, 18, 46656, 6480, 216, 1, 279936, 46656, 2160, 24, 1679616, 326592, 19440, 360, 1, 10077696, 2239488, 163296, 4320, 30, 60466176, 15116544, 1306368, 45360, 540, 1, 362797056, 100776960, 10077696, 435456, 7560, 36

Offset: 0

Zagros Lalo, May 09 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013613 ((1+6*x)^n).
The coefficients in the expansion of 1/(1-6x-x^2) are given by the sequence generated by the row sums.
The row sums are Denominators of continued fraction convergent to sqrt(10), see A005668.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 6.162277660..., a metallic mean (see A176398), when n approaches infinity.

			Triangle begins:
1;
6;
36, 1;
216, 12;
1296, 108, 1;
7776, 864, 18;
46656, 6480, 216, 1;
279936, 46656, 2160, 24;
1679616, 326592, 19440, 360, 1;
10077696, 2239488, 163296, 4320, 30;
60466176, 15116544, 1306368, 45360, 540, 1;
362797056, 100776960, 10077696, 435456, 7560, 36;
2176782336, 665127936, 75582720, 3919104, 90720, 756, 1;
13060694016, 4353564672, 554273280, 33592320, 979776, 12096, 42;
78364164096, 28298170368, 3990767616, 277136640, 9797760, 163296, 1008, 1;
470184984576, 182849716224, 28298170368, 2217093120, 92378880, 1959552, 18144, 48;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 94.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1+6x)^n

Row sums give A005668.
Cf. A000400 (column 0), A053469 (column 1), A081136 (column 2), A081144 (column 3).
Cf. A013613.
Cf. A176398.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 6 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, 6*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, May 26 2018

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

Original entry on oeis.org

1, -3, 4, 9, -32, 25, -27, 192, -375, 216, 81, -1024, 3750, -5184, 2401, -243, 5120, -31250, 77760, -84035, 32768, 729, -24576, 234375, -933120, 1764735, -1572864, 531441, -2187, 114688, -1640625, 9797760, -28824005, 44040192, -33480783, 10000000, 6561, -524288, 10937500, -94058496, 403536070, -939524096, 1205308188, -800000000, 214358881

Offset: 0

Harlan J. Brothers and Wolfdieter Lang, Apr 27 2023

This is the member N = 2 of a family of signed triangles with row sums n! = A000142(n): T(N; n, k) = (-1)^(n-k)*binomial(n, k)*(k + N + 1)^n, for integer N, n >= 0 and k = 0, 1, ..., n. The row polynomials PS(N; n, z) = Sum_{k=0..n} T(N; n, k)*z^k = ((-1)^n/z^N)*D_{n,N+1,n}(z) in [Sidi 1980].
For N = -1, 0 and 1 see A258773(n, k), A075513(n+1, k) and (-1)^(n-k) * A154715(n, k), respectively.
The column sequences, for k = 0, 1, ..., 6 and n >= k, are A141413(n+2), (-1)^(n+1)*A018215(n) = 4*(-1)^(n+1)*A002697(n), 5^2*(-1)^n*A081135(n), (-1)^(n+1)*A128964(n-1) = 6^3*(-1)^(n+1)*A081144(n), 7^4*(-1)^n*A139641(n-4), 2^15*(-1)^(n+1)*A173155(n-5), 3^12*(-1)^n*A173191(n-6), respectively.
The e.g.f. of the triangle (see below) needs the exponential convolution (LambertW(-z)/(-z))^2 = Sum_{n>=0} c(2; n)*z^n/n!, where c(2; n) = Sum_{m=0..n} |A137352(n+1, m)|*2^m = A007334(n+2).
The row sums give n! = A000142(n).

			The triangle T begins:
n\k    0       1        2         3         4          5          6         7
0:     1
1:    -3       4
2:     9     -32       25
3:   -27     192     -375       216
4:    81   -1024     3750     -5184      2401
5:  -243    5120   -31250     77760    -84035      32768
6:   729  -24576   234375   -933120   1764735   -1572864     531441
7: -2187  114688 -1640625   9797760 -28824005   44040192  -33480783  10000000
...
n = 8:  6561 -524288 10937500 -94058496 403536070 -939524096 1205308188 -800000000 2143588,
n = 9: -19683 2359296 -70312500 846526464 -5084554482 16911433728 -32543321076 36000000000 -21221529219 5159780352.

Paolo Xausa, Table of n, a(n) for n = 0..5049 (rows 0..100 of the triangle, flattened)
Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Cf. A000142 (row sums), A075513, A154715, A258773.

Columns k = 0..6 involve (see above): A002697, A007334, A018215, A081135, A081144, A128964, A137352, A139641, A141413, A173155, A173191.

A362353row[n_]:=Table[(-1)^(n-k)Binomial[n,k](k+3)^n,{k,0,n}];Array[A362353row,10,0] (* Paolo Xausa, Jul 30 2023 *)

T(n, k) = (-1)^(n-k)*binomial(n, k)*(k + 3)^n, for n >= 0, k = 0, 1, ..., n.
O.g.f. of column k: (x*(k + 3))^k/(1 - (k + 3)*x)^(k+1), for k >= 0.
E.g.f. of column k: exp(-(k + 3)*x)*((k + 3)*x)^k/k!, for k >= 0.
E.g.f. of the triangle, that is, the e.g.f. of its row polynomials {PS(2;n,y)}_{n>=0}): ES(2;y,x) = exp(-3*x)*(1/3)*(d/dz)(W(-z)/(-z))^2, after replacing z by x*y*exp(-x), where W is the Lambert W-function for the principal branch. This becomes ES(2;y,x) = exp(-3*x)*exp(3*(-W(-z)))/(1 - (-W(-z)), with z = x*y*exp(-x).

a(41)-a(44) from Paolo Xausa, Jul 31 2023

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A128964 a(n) = (n^3-n)*6^n.

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

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A172501 a(n) = binomial(n+8,8)*6^n.

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A173123 a(n) = binomial(n+9,9)*6^n.

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A173124 a(n) = binomial(n+10,10)*6^n.

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A304255 Triangle read by rows: T(0,0) = 1; T(n,k) = 6*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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A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)*binomial(n, k)*(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

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A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.