A017933 -id:A017933 - cp's OEIS frontend

A075498 Stirling2 triangle with scaled diagonals (powers of 3).

1, 3, 1, 9, 9, 1, 27, 63, 18, 1, 81, 405, 225, 30, 1, 243, 2511, 2430, 585, 45, 1, 729, 15309, 24381, 9450, 1260, 63, 1, 2187, 92583, 234738, 137781, 28350, 2394, 84, 1, 6561, 557685, 2205225, 1888110, 563031, 71442, 4158, 108, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(3*z) - 1)*x/3) - 1.
Subtriangle of the triangle given by (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938, see example. - Philippe Deléham, Feb 13 2013
Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			[1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
From _Philippe Deléham_, Feb 13 2013: (Start)
Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1;
  0,   1;
  0,   3,   1;
  0,   9,   9,   1;
  0,  27,  63,  18,   1;
  0,  81, 405, 225,  30,   1;
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212.

Cf. A075497, A075499.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 3^n, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
rows = 9;
t = Table[3^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (3^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 3*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
From Peter Bala, Jan 13 2018: (Start)
Dobinski-type formulas for row polynomials R(n,x):
R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)

A016137 Expansion of 1/((1-3x)(1-6*x)).

Original entry on oeis.org

1, 9, 63, 405, 2511, 15309, 92583, 557685, 3352671, 20135709, 120873303, 725416965, 4353033231, 26119793709, 156723545223, 940355620245, 5642176768191, 33853189749309, 203119525916343, 1218718317759525, 7312313393341551, 43873890820402509, 263243376303474663, 1579460351964026805, 9476762394213697311

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-18).

Second column of triangle A075498.

Cf. A000225, A000244, A001045, A008277, A016127, A016129, A017933, A100852.

[2*6^n -3^n: n in [0..40]]; // G. C. Greubel, Nov 14 2024

Table[2*6^n -3^n, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
CoefficientList[Series[1/((1-3x)(1-6x)),{x,0,40}],x] (* or *) LinearRecurrence[{9,-18},{1,9},40] (* Harvey P. Dale, Jul 07 2012 *)

Vec(1/(1-3*x)/(1-6*x)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

[lucas_number1(n,9,18) for n in range(1,41)] # Zerinvary Lajos, Apr 23 2009

a(n) = (3^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = 2*6^n - 3^n.
E.g.f.: (d^2/dx^2)((((exp(3*x)-1)/3)^2)/2!) = -exp(3*x) + 2*exp(6*x).
With leading zero, this is (6^n - 3^n)/3, the binomial transform of A016127 (with extra leading zero). - Paul Barry, Aug 20 2003
With leading zero, this is the fourth binomial transform of A001045, with a(n) = (2^n-1)(3^n/3 - 0^n/3) = A000225(n)*(A000244(n-1) - 0^n/3). - Paul Barry, Apr 28 2004
a(n) = Sum_{k=0..n} A100852(n,k). - Reinhard Zumkeller, Nov 20 2004
Sum_{k=1..n} 3^(k-1)*3^(n-k)*binomial(n, k). - Zerinvary Lajos, Sep 24 2006
a(n) = 9*a(n-1) - 18*a(n-2), n >= 2. - Vincenzo Librandi, Mar 14 2011

More terms added by G. C. Greubel, Nov 14 2024

A028085 Expansion of 1/((1-3x)(1-6x)(1-9x)(1-12x)).

Original entry on oeis.org

1, 30, 585, 9450, 137781, 1888110, 24862545, 318755250, 4012058061, 49847787990, 613622150505, 7503229474650, 91300979746341, 1106997911204670, 13386607046238465, 161563913916523650

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (30,-315,1350,-1944).

Fourth column of triangle A075498.

Cf. A017933, A075515.

CoefficientList[Series[1/((1-3x)(1-6x)(1-9x)(1-12x)),{x,0,30}],x] (* or *) LinearRecurrence[{30,-315,1350,-1944},{1,30,585,9450},30] (* Harvey P. Dale, Feb 06 2015 *)

Vec(1/((1-3*x)*(1-6*x)*(1-9*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (3^n)*Stirling2(n+4, 4), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = Sum_{m=0..3} (A075513(4, m)*((m+1)*3)^n)/3!.
G.f.: 1/Product_{k=1..4} (1-3*k*x).
E.g.f.: (d^4/dx^4)((((exp(3*x)-1)/3)^4)/4!) = Sum_{m=0..3} (A075513(4, m)*exp(3*(m+1)*x))/3!.
a(n) = (12^(n+3) - 3*9^(n+3) + 3*6^(n+3) - 3^(n+3))/162. - Yahia Kahloune, Jun 10 2013
a(0)=1, a(1)=30, a(2)=585, a(3)=9450, a(n) = 30*a(n-1) - 315*a(n-2) + 1350*a(n-3) - 1944*a(n-4). - Harvey P. Dale, Feb 06 2015

A019618 Expansion of 1/((1-4x)(1-7x)(1-10*x)).

Original entry on oeis.org

1, 21, 303, 3745, 42711, 464961, 4918663, 51086385, 524227671, 5336085601, 54018566823, 544793838225, 5480212349431, 55028108373441, 551863246323783, 5529708675105265, 55374624529091991, 554289026917064481, 5546689809273133543, 55493495148326663505, 555121131971945559351

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (21,-138,280).

Cf. A016223, A016297, A017933, A020447.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-7*x)*(1-10*x)))); // Vincenzo Librandi, Jul 03 2013

I:=[1, 21, 303]; [n le 3 select I[n] else 21*Self(n-1)-138*Self(n-2)+280*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 7 x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{21,-138,280},{1,21,303},30] (* Harvey P. Dale, Mar 09 2017 *)

x='x+O('x^30); Vec(1/((1-4*x)*(1-7*x)*(1-10*x))) \\ G. C. Greubel, Aug 24 2018

a(n) = (2*4^(n+1) -7^(n+2) +5*10^(n+1))/9. - R. J. Mathar, Nov 11 2012
a(0)=1, a(1)=21, a(2)=303; for n>2, a(n) = 21*a(n-1) -138*a(n-2) +280*a(n-3). - Vincenzo Librandi, Jul 03 2013
a(n) = 17*a(n-1) -70*a(n-2) +4^n. - Vincenzo Librandi, Jul 03 2013
From Seiichi Manyama, May 05 2025: (Start)
a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-3)^k * 10^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

A020447 Expansion of 1/((1-5x) (1-8x) (1-11*x)).

Original entry on oeis.org

1, 24, 393, 5480, 70161, 853944, 10066393, 116192520, 1322205921, 14898923864, 166735197993, 1856912289960, 20608880226481, 228161663489784, 2521496249891193, 27830232878409800, 306882907287251841, 3381715508097175704, 37246902627265441993, 410100204278978264040

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (24,-183,440).

Cf. A016223, A016297, A017933, A019618.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-5*x)*(1-8*x)*(1-11*x)))); // Vincenzo Librandi, Jul 03 2013

I:=[1, 24, 393]; [n le 3 select I[n] else 24*Self(n-1)-183*Self(n-2)+440*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013

CoefficientList[Series[1 / ((1 - 5 x) (1 - 8 x) (1 - 11 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{24,-183,440},{1,24,393},30] (* Harvey P. Dale, Jun 20 2015 *)

a(n) = 25*5^n/18 -64*8^n/9 +121*11^n/18. - R. J. Mathar, Jun 29 2013
a(0)=1, a(1)=24, a(2)=393; for n>2, a(n) = 24*a(n-1) -183*a(n-2) +440*a(n-3). - Vincenzo Librandi, Jul 03 2013
a(n) = 19*a(n-1) -88*a(n-2) +5^n. - Vincenzo Librandi, Jul 03 2013
From Seiichi Manyama, May 05 2025: (Start)
a(n) = Sum_{k=0..n} 3^k * 5^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-3)^k * 11^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

A020724 G.f.: 1/((1-6x)(1-9x)(1-12*x)).

Original entry on oeis.org

1, 27, 495, 7695, 109431, 1474767, 19198215, 244167615, 3055792311, 37811613807, 464078786535, 5662363023135, 68791291322391, 833094971965647, 10065613312624455, 121404092777808255, 1462402531680439671

Offset: 0

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 0..925
Index entries for linear recurrences with constant coefficients, signature (27,-234,648).

Cf. A017933.

CoefficientList[Series[1/((1-6x)(1-9x)(1-12x)),{x,0,20}],x] (* or *) LinearRecurrence[{27,-234,648},{1,27,495},20] (* Harvey P. Dale, Feb 18 2025 *)

Vec(1/((1-6*x)*(1-9*x)*(1-12*x)) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

a(n) = (12^(n+2) - 2*9^(n+2) + 6^(n+2))/18. - Yahia Kahloune, Jun 30 2013

a(n) = 27*a(n-1) - 234*a(n-2) + 648*a(n-3). - Jinyuan Wang, Mar 10 2020

cp's OEIS Frontend

A075498 Stirling2 triangle with scaled diagonals (powers of 3).

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A016137 Expansion of 1/((1-3*x)*(1-6*x)).

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A028085 Expansion of 1/((1-3x)(1-6x)(1-9x)(1-12x)).

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A019618 Expansion of 1/((1-4*x)*(1-7*x)*(1-10*x)).

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A020447 Expansion of 1/((1-5*x) * (1-8*x) * (1-11*x)).

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Magma

Magma

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A020724 G.f.: 1/((1-6*x)*(1-9*x)*(1-12*x)).

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A016137 Expansion of 1/((1-3x)(1-6*x)).

A019618 Expansion of 1/((1-4x)(1-7x)(1-10*x)).

A020447 Expansion of 1/((1-5x) (1-8x) (1-11*x)).

A020724 G.f.: 1/((1-6x)(1-9x)(1-12*x)).