A172362 -id:A172362 - cp's OEIS frontend

A036216 Expansion of 1/(1 - 3*x)^4; 4-fold convolution of A000244 (powers of 3).

1, 12, 90, 540, 2835, 13608, 61236, 262440, 1082565, 4330260, 16888014, 64481508, 241805655, 892820880, 3252418920, 11708708112, 41712272649, 147219785820, 515269250370, 1789882659180, 6175095174171, 21171754882872

Offset: 0

Wolfdieter Lang

With three leading zeros, 3rd binomial transform of (0,0,0,1,0,0,0,0,...). - Paul Barry, Mar 07 2003

Number of n-permutations (n=4) of 4 objects u, v, w, z, with repetition allowed, containing exactly three u's. - Zerinvary Lajos, May 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Cf. A027465.

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

[3^n* Binomial(n+3, 3): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

seq(3^n*binomial(n+3, 3), n=0..30)]; # Zerinvary Lajos, Dec 21 2006

CoefficientList[Series[1/(1-3x)^4,{x,0,30}],x] (* or *) LinearRecurrence[ {12,-54,108,-81},{1,12,90,540},30] (* Harvey P. Dale, Jul 27 2017 *)

a(n) = 3^n*binomial(n+3, 3) \\ Charles R Greathouse IV, Oct 03 2016

[3^n*binomial(n+3,3) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

a(n) = 3^n*binomial(n+3, 3).
a(n) = A027465(n+4, 4).
G.f.: 1/(1 - 3*x)^4.
With three leading zeros, a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4), a(0) = a(1) = a(2) = 0, a(3) = 1. - Paul Barry, Mar 07 2003
With three leading zeros, C(n, 3)*3^(n-3) is the second binomial transform of C(n, 3). - Paul Barry, Jul 24 2003
E.g.f.: (1/2)*(2 + 18*x + 27*x^2 + 9*x^3)*exp(3*x). - Franck Maminirina Ramaharo, Nov 23 2018
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 36*log(3/2) - 27/2.
Sum_{n>=0} (-1)^n/a(n) = 144*log(4/3) - 81/2. (End)

A013610 Triangle of coefficients in expansion of (1+3*x)^n.

Original entry on oeis.org

1, 1, 3, 1, 6, 9, 1, 9, 27, 27, 1, 12, 54, 108, 81, 1, 15, 90, 270, 405, 243, 1, 18, 135, 540, 1215, 1458, 729, 1, 21, 189, 945, 2835, 5103, 5103, 2187, 1, 24, 252, 1512, 5670, 13608, 20412, 17496, 6561, 1, 27, 324, 2268, 10206, 30618, 61236, 78732, 59049, 19683

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,k) with steps (1,0) and three kinds of steps (1,1). The number of paths with steps (1,0) and s kinds of steps (1,1) corresponds to the expansion of (1+s*x)^n. - Joerg Arndt, Jul 01 2011
Rows of A027465 reversed. - Michael Somos, Feb 14 2002
T(n,k) equals the number of n-length words on {0,1,2,3} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 3-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference.  - Brian Hopkins, Aug 16 2020

			Triangle begins
  1;
  1,    3;
  1,    6,    9;
  1,    9,   27,   27;
  1,   12,   54,  108,   81;
  1,   15,   90,  270,  405,  243;
  1,   18,  135,  540, 1215, 1458,  729;
  1,   21,  189,  945, 2835, 5103, 5103, 2187;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients on the 3xn board (all heights 3) with k rooks
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.

Cf. A007318, A013609, A027465, etc.

Diagonals of the triangle: A000244 (k=n), A027471 (k=n-1), A027472 (k=n-2), A036216 (k=n-3), A036217 (k=n-4), A036219 (k=n-5), A036220 (k=n-6), A036221 (k=n-7), A036222 (k=n-8), A036223 (k=n-9), A172362 (k=n-10).

a013610 n k = a013610_tabl !! n !! k
a013610_row n = a013610_tabl !! n
a013610_tabl = iterate (\row ->
   zipWith (+) (map (* 1) (row ++ [0])) (map (* 3) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013

[3^k*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 19 2021

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+3*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 25 2015

t[n_, k_] := Binomial[n, k]*3^(n-k); Table[t[n, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2013 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 4], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)
T[0, 0] := 1; T[n_, k_]/;0<=k<=n := T[n, k] = 3T[n-1, k-1]+T[n-1, k]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Oliver Seipel, Jan 26 2025 *)

{T(n, k) = polcoeff((1 + 3*x)^n, k)}; /* Michael Somos, Feb 14 2002 */

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,1], [1,1]]; /* note triple [1,1] */
/* Joerg Arndt, Jul 01 2011 */

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

G.f.: 1 / (1 - x*(1+3*y)).
Row sums are 4^n. - Joerg Arndt, Jul 01 2011
T(n,k) = 3^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*2^(n-i). - Mircea Merca, Apr 28 2012
From Peter Bala, Dec 22 2014: (Start)
Riordan array ( 1/(1 - x), 3*x/(1 - x) ).
exp(3*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(3*x)*(1 + 9*x + 27*x^2/2! + 27*x^3/3!) = 1 + 12*x + 90*x^2/2! + 540*x^3/3! + 2835*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 3*x/(1 - x) ). (End)
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 4^j. - Kolosov Petro, Jan 28 2019
T(0,0)=1, T(n,k)=3*T(n-1,k-1)+T(n-1,k) for 0<=k<=n, T(n,k)=0 for k<0 or k>n. - Oliver Seipel, Feb 10 2025

A036217 Expansion of 1/(1-3*x)^5; 5-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 15, 135, 945, 5670, 30618, 153090, 721710, 3247695, 14073345, 59108049, 241805655, 967222620, 3794488740, 14635885140, 55616363532, 208561363245, 772903875555, 2833980877035, 10291825290285, 37050571045026

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n=5) of 4 objects: u, v, z, x with repetition allowed, containing exactly four (4) u's. Example: a(1)=15 because we have uuuuv uuuvu uuvuu uvuuu vuuuu uuuuz uuuzu uuzuu uzuuu zuuuu uuuux uuuxu uuxuu uxuuu xuuuu. - Zerinvary Lajos, Jun 12 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,243).

Cf. A000332, A027465.

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

[3^n* Binomial(n+4, 4): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

seq(3^n*binomial(n+4,4), n=0..30); # Zerinvary Lajos, Jun 12 2008

CoefficientList[Series[1/(1-3x)^5,{x,0,30}],x] (* Harvey P. Dale, Jun 13 2017 *)

[3^n*binomial(n+4,4) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

a(n) = 3^n*binomial(n+4, 4) = 3^n*A000332(n+4).
a(n) = A027465(n+5, 5).
G.f.: 1/(1-3*x)^5.
E.g.f.: (1/8)*(8 +96*x +216*x^2 +144*x^3 +27*x^4)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 40 - 96*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 768*log(4/3) - 220. (End)

A036219 Expansion of 1/(1-3*x)^6; 6-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 18, 189, 1512, 10206, 61236, 336798, 1732104, 8444007, 39405366, 177324147, 773778096, 3288556908, 13660159464, 55616363532, 222465454128, 875957725629, 3400777052442, 13036312034361, 49400761393368, 185252855225130, 688082033693340, 2533392942234570

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (18,-135,540,-1215,1458,-729).

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), this sequence (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

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

seq(3^n*binomial(n+5,5), n=0..30); # Zerinvary Lajos, Jun 13 2008

Table[3^n*Binomial[n+5, 5], {n, 0, 30}] (* G. C. Greubel, May 19 2021 *)
CoefficientList[Series[1/(1-3x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {18,-135,540,-1215,1458,-729},{1,18,189,1512,10206,61236},30] (* Harvey P. Dale, Jan 02 2022 *)

[3^n*binomial(n+5,5) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

a(n) = 3^n*binomial(n+5, 5).
a(n) = A027465(n+6, 6).
G.f.: 1/(1-3*x)^6.
E.g.f.: (1/40)*(40 + 600*x + 1800*x^2 + 1800*x^3 + 675*x^4 + 81*x^5)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 240*log(3/2) - 385/4.
Sum_{n>=0} (-1)^n/a(n) = 3840*log(4/3) - 4415/4. (End)

A036220 Expansion of 1/(1-3*x)^7; 7-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 21, 252, 2268, 17010, 112266, 673596, 3752892, 19702683, 98513415, 472864392, 2192371272, 9865670724, 43257171636, 185387878440, 778629089448, 3211844993973, 13036312034361, 52145248137444, 205836505805700, 802762372642230, 3096369151620030

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (21,-189,945,-2835,5103,-5103,2187).

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), this sequence (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

[3^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+6,6), n=0..20); # Zerinvary Lajos, Jun 16 2008

Table[3^n*Binomial[n+6, 6], {n,0,30}] (* G. C. Greubel, May 19 2021 *)

[3^n*binomial(n+6,6) for n in range(30)] # Zerinvary Lajos, Mar 10 2009

a(n) = 3^n*binomial(n+6, 6).
a(n) = A027465(n+7,7).
G.f.: 1/(1-3*x)^7.
E.g.f.: (1/80)*(80 + 1440*x + 5400*x^2 + 7200*x^3 + 4050*x^4 + 972*x^5 + 81*x^6)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 1173/5 - 576*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 18432*log(4/3) - 26508/5. (End)

A036221 Expansion of 1/(1-3*x)^8; 8-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 24, 324, 3240, 26730, 192456, 1250964, 7505784, 42220035, 225173520, 1148384952, 5637526128, 26778249108, 123591918960, 556163635320, 2447119995408, 10553204980197, 44695926974952, 186233029062300

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n>=7) of 4 objects: u, v, z, x with repetition allowed, containing exactly seven (7) u's. Example: a(1)=24 because we have uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu. - Zerinvary Lajos, Jun 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-252,1512,-5670,13608,-20412,17496,-6561).

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), this sequence (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

[3^n*Binomial(n+7, 7): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+7,7), n=0..30); # Zerinvary Lajos, Jun 23 2008

Table[3^n*Binomial[n+7,7], {n,0,30}] (* G. C. Greubel, May 19 2021 *)

[3^n*binomial(n+7, 7) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

a(n) = 3^n*binomial(n+7, 7).
a(n) = A027465(n+8, 8.)
G.f.: 1/(1-3*x)^8.
E.g.f.: (1/560)*(560 +11760*x +52920*x^2 +88200*x^3 +66150*x^4 +23814*x^5 +3969*x^6 +243*x^7)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 1344*log(3/2) - 5439/10.
Sum_{n>=0} (-1)^n/a(n) = 86016*log(4/3) - 247443/10. (End)

A036222 Expansion of 1/(1-3*x)^9; 9-fold convolution of A000244 (powers of 3).

Original entry on oeis.org

1, 27, 405, 4455, 40095, 312741, 2189187, 14073345, 84440070, 478493730, 2583866142, 13389124554, 66945622770, 324428787270, 1529449997130, 7035469986798, 31659614940591, 139674771796725, 605257344452475

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n>=8) of 4 objects: u, v, z, x with repetition allowed, containing exactly eight (8) u's. Example: a(1)=27 because we have uuuuuuuuv, uuuuuuuuz, uuuuuuuux, uuuuuuuvu, uuuuuuuzu, uuuuuuuxu, uuuuuuvuu, uuuuuuzuu, uuuuuuxuu, uuuuuvuuu, uuuuuzuuu, uuuuuxuuu, uuuuvuuuu, uuuuzuuuu, uuuuxuuuu, uuuvuuuuu, uuuzuuuuu, uuuxuuuuu, uuvuuuuuu, uuzuuuuuu, uuxuuuuuu, uvuuuuuuu, uzuuuuuuu, uxuuuuuuu, vuuuuuuuu, zuuuuuuuu, xuuuuuuuu. - Zerinvary Lajos, Jun 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (27,-324,2268,-10206,30618,-61236,78732,-59049,19683).

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), this sequence (m=8), A036223 (m=9), A172362 (m=10).

[3^n*Binomial(n+8, 8): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(3^n*binomial(n+8,8), n=0..18); # Zerinvary Lajos, Jun 23 2008

Table[3^n*Binomial[n+8, 8], {n, 0, 20}] (* Zerinvary Lajos, Jan 31 2010 *)
CoefficientList[Series[1/(1-3x)^9,{x,0,30}],x] (* or *) LinearRecurrence[{27,-324, 2268,-10206,30618,-61236,78732,-59049,19683}, {1,27,405,4455,40095,312741, 2189187,14073345,84440070}, 30] (* Harvey P. Dale, Jan 07 2016 *)

[3^n*binomial(n+8, 8) for n in range(30)] # Zerinvary Lajos, Mar 13 2009

a(n) = 3^n*binomial(n+8, 8).
a(n) = A027465(n+9, 9).
G.f.: 1/(1-3*x)^9.
a(0)=1, a(1)=27, a(2)=405, a(3)=4455, a(4)=40095, a(5)=312741, a(6)=2189187, a(7)=14073345, a(8)=84440070, a(n) = 27*a(n-1) - 324*a(n-2) + 2268*a(n-3) - 10206*a(n-4) + 30618*a(n-5) - 61236*a(n-6) + 78732*a(n-7) - 59049*a(n-8) + 19683*a(n-9). - Harvey P. Dale, Jan 07 2016
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 43632/35 - 3072*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 393216*log(4/3) - 3959208/35. (End)

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

Original entry on oeis.org

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)

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

cp's OEIS Frontend

A036216 Expansion of 1/(1 - 3*x)^4; 4-fold convolution of A000244 (powers of 3).

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A013610 Triangle of coefficients in expansion of (1+3*x)^n.

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A036217 Expansion of 1/(1-3*x)^5; 5-fold convolution of A000244 (powers of 3).

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A036219 Expansion of 1/(1-3*x)^6; 6-fold convolution of A000244 (powers of 3).

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A036220 Expansion of 1/(1-3*x)^7; 7-fold convolution of A000244 (powers of 3).

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A036221 Expansion of 1/(1-3*x)^8; 8-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.