A013620
Triangle of coefficients in expansion of (2+3x)^n.
Original entry on oeis.org
1, 2, 3, 4, 12, 9, 8, 36, 54, 27, 16, 96, 216, 216, 81, 32, 240, 720, 1080, 810, 243, 64, 576, 2160, 4320, 4860, 2916, 729, 128, 1344, 6048, 15120, 22680, 20412, 10206, 2187, 256, 3072, 16128, 48384, 90720, 108864, 81648, 34992, 6561, 512
Offset: 0
Triangle begins:
1;
2,3;
4,12,9;
8,36,54,27;
16,96,216,216,81;
-
a013620 n k = a013620_tabl !! n !! k
a013620_row n = a013620_tabl !! n
a013620_tabl = iterate (\row ->
zipWith (+) (map (* 2) (row ++ [0])) (map (* 3) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011
-
Flatten[Table[Binomial[i, j] 2^(i-j) 3^j, {i, 0, 10}, {j, 0, i}]] (* Vincenzo Librandi, Apr 22 2014 *)
A038220
Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
Original entry on oeis.org
1, 3, 2, 9, 12, 4, 27, 54, 36, 8, 81, 216, 216, 96, 16, 243, 810, 1080, 720, 240, 32, 729, 2916, 4860, 4320, 2160, 576, 64, 2187, 10206, 20412, 22680, 15120, 6048, 1344, 128, 6561, 34992, 81648, 108864, 90720, 48384, 16128, 3072, 256
Offset: 0
Triangle begins:
1;
3, 2;
9, 12, 4;
27, 54, 36, 8;
81, 216, 216, 96, 16;
...
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48
-
a038220 n k = a038220_tabl !! n !! k
a038220_row n = a038220_tabl !! n
a038220_tabl = iterate (\row ->
zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011
-
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 t[n - 1, k] + 2 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)
-
T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ Charles R Greathouse IV, Jul 19 2016
A154692
Triangle read by rows: T(n, k) = (2^(n-k)*3^k + 2^k*3^(n-k))*binomial(n, k).
Original entry on oeis.org
2, 5, 5, 13, 24, 13, 35, 90, 90, 35, 97, 312, 432, 312, 97, 275, 1050, 1800, 1800, 1050, 275, 793, 3492, 7020, 8640, 7020, 3492, 793, 2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315, 6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817
Offset: 0
Triangle begins
2;
5, 5;
13, 24, 13;
35, 90, 90, 35;
97, 312, 432, 312, 97;
275, 1050, 1800, 1800, 1050, 275;
793, 3492, 7020, 8640, 7020, 3492, 793;
2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315;
6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817;
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
- A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.
-
A154692:= func< n,k | (2^(n-k)*3^k + 2^k*3^(n-k))*Binomial(n,k) >;
[A154692(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025
-
A154692 := proc(n,m)
(2^(n-m)*3^m+2^m*3^(n-m))*binomial(n,m) ;
end proc:
seq(seq(A154692(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Oct 24 2011
-
p=2; q=3;
T[n_, m_]= (p^(n-m)*q^m + p^m*q^(n-m))*Binomial[n,m];
Table[T[n,m], {n,0,10}, {m,0,n}]//Flatten
-
from sage.all import *
def A154692(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*binomial(n,k)
print(flatten([[A154692(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025
Showing 1-3 of 3 results.
Comments