A036216 -id:A036216 - cp's OEIS frontend

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.

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).

A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 24, 4, 0, 1, 20, 90, 80, 5, 0, 1, 30, 240, 540, 240, 6, 0, 1, 42, 525, 2240, 2835, 672, 7, 0, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 0, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 0, 1, 90, 2880, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1,
1,  0,
1,  2,   0,
1,  6,   3,    0,
1, 12,  24,    4,    0,
1, 20,  90,   80,    5,   0,
1, 30, 240,  540,  240,   6, 0,
1, 42, 525, 2240, 2835, 672, 7, 0,
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

There are 4 versions: A059297-A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.

/* As triangle: */ [[Binomial(n,k)*(n-k)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

T := (n, k) -> binomial(n, k) * (n - k)^k:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

t[n_, k_] := Binomial[n, k]*(n - k)^k; Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

concat([1], for(n=0, 25, for(k=0, n, print1(binomial(n,k)*(n-k)^k, ", ")))) \\ G. C. Greubel, Jan 05 2017

Name corrected by Peter Luschny, Nov 12 2023

A217629 Triangle, read by rows, where T(n,k) = k!C(n, k)3^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 3, 1, 9, 6, 2, 27, 27, 18, 6, 81, 108, 108, 72, 24, 243, 405, 540, 540, 360, 120, 729, 1458, 2430, 3240, 3240, 2160, 720, 2187, 5103, 10206, 17010, 22680, 22680, 15120, 5040, 6561, 17496, 40824, 81648, 136080, 181440, 181440, 120960, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=3.
Sum(T(n,k), k=0..n) = A053486(n) (see the Formula section of A053486). Also:
first column:     A000244;
second column:    A027471;
third column:   2*A027472;
fourth column:  6*A036216;
fifth column:  24*A036217.

			Triangle begins:
1;
3,     1;
9,     6,     2;
27,    27,    18,     6;
81,    108,   108,    72,     24;
243,   405,   540,    540,    360,    120;
729,   1458,  2430,   3240,   3240,   2160,    720;
2187,  5103,  10206,  17010,  22680,  22680,   15120,   5040;
6561,  17496, 40824,  81648,  136080, 181440,  181440,  120960,  40320; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000244, A027471, A027472, A036216, A036217, A053486, A090802, A218016, A218017.

[Factorial(n)/Factorial(n-k)*3^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*3^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 3^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(3x)*x^k.

A318773 Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), 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, 81, 1, 243, 6, 729, 27, 2187, 108, 6561, 405, 1, 19683, 1458, 9, 59049, 5103, 54, 177147, 17496, 270, 531441, 59049, 1215, 1, 1594323, 196830, 5103, 12, 4782969, 649539, 20412, 90, 14348907, 2125764, 78732, 540, 43046721, 6908733, 295245, 2835, 1, 129140163, 22320522, 1082565, 13608, 15

Offset: 0

Zagros Lalo, Sep 04 2018

The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in center-justified 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/(1-3*x-x^4) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.035744112294..., when n approaches infinity.

			Triangle begins:
          1;
          3;
          9;
         27;
         81,        1;
        243,        6;
        729,       27;
       2187,      108;
       6561,      405,       1;
      19683,     1458,       9;
      59049,     5103,      54;
     177147,    17496,     270;
     531441,    59049,    1215,     1;
    1594323,   196830,    5103,    12;
    4782969,   649539,   20412,    90;
   14348907,  2125764,   78732,   540;
   43046721,  6908733,  295245,  2835,   1;
  129140163, 22320522, 1082565, 13608,  15;
  387420489, 71744535, 3897234, 61236, 135;
  ...

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

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

Row sums give A052917.
Cf. A013610, A027465.
Cf. A304236, A304249
Cf. A317496, A317497.
Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3).
Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), A304249 (q=2), A317497 (q=3), this sequence (q=4).

[3^(n-4*k)*Binomial(n-3*k,k): k in [0..Floor(n/4)], n in [0..24]]; // G. C. Greubel, May 12 2021

T[n_, k_]:= T[n, k] = 3^(n-4k)*(n-3k)!/((n-4k)! k!); Table[T[n, k], {n, 0, 17}, {k, 0, Floor[n/4]} ]//Flatten
T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, 3T[n-1, k] + T[n-4, k-1]]; Table[T[n, k], {n, 0, 17}, {k, 0, Floor[n/4]}]//Flatten

flatten([[3^(n-4*k)*binomial(n-3*k,k) for k in (0..n//4)] for n in (0..24)]) # G. C. Greubel, May 12 2021

T(n,k) = 3^(n-4*k) * (n-3*k)!/(k! * (n-4*k)!) where n >= 0 and 0 <= k <= floor(n/4).

A129003 a(n) = (n^3 + n^2)*3^n.

Original entry on oeis.org

6, 108, 972, 6480, 36450, 183708, 857304, 3779136, 15943230, 64953900, 257217444, 994857552, 3772168218, 14061928860, 51656065200, 187339329792, 671787127926, 2384960530284, 8391527791740, 29288988968400, 101486346775506

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 (12,-54,108,-81).

Cf. A036216, A128796, A036289.

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

I:=[6,108,972,6480]; [n le 4 select I[n] else 12*Self(n-1)-54*Self(n-2)+108*Self(n-3)-81*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

LinearRecurrence[{12, -54, 108, -81}, {6, 108, 972, 6480}, 30] (* Vincenzo Librandi, Feb 12 2013 *)

G.f.: 6*x*(1+6*x)/(1-3*x)^4. - R. J. Mathar, Dec 19 2008
a(n) = 12*a(n-1)-54*a(n-2)+108*a(n-3)-81*a(n-4). - Vincenzo Librandi, Feb 12 2013
a(n) = 6*(A036216(n-1)+6*A036216(n-2)), with A036216(-1)=0. - Bruno Berselli, Feb 12 2013

cp's OEIS Frontend

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.

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A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

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A217629 Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.

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A318773 Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

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A129003 a(n) = (n^3 + n^2)*3^n.

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A217629 Triangle, read by rows, where T(n,k) = k!C(n, k)3^(n-k) for n>=0, k=0..n.