A077385 Triangle read by rows in which n-th row contains 2n-1 terms starting from n^0 to n^(n-1) in increasing order and then in decreasing order to n^0.
1, 1, 2, 1, 1, 3, 9, 3, 1, 1, 4, 16, 64, 16, 4, 1, 1, 5, 25, 125, 625, 125, 25, 5, 1, 1, 6, 36, 216, 1296, 7776, 1296, 216, 36, 6, 1, 1, 7, 49, 343, 2401, 16807, 117649, 16807, 2401, 343, 49, 7, 1, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 262144, 32768, 4096, 512, 64, 8, 1
Offset: 1
Examples
Irregular triangle begins as: 1; 1, 2, 1; 1, 3, 9, 3, 1; 1, 4, 16, 64, 16, 4, 1; 1, 5, 25, 125, 625, 125, 25, 5, 1; 1, 6, 36, 216, 1296, 7776, 1296, 216, 36, 6, 1; 1, 7, 49, 343, 2401, 16807, 117649, 16807, 2401, 343, 49, 7, 1;
Links
- G. C. Greubel, Rows n = 1..40 of the irregular triangle, flattened
Programs
-
Magma
A077385:= func< n,k | k lt n select n^k else n^(2*n-k-2) >; [A077385(n,k): k in [0..2*n-2], n in [1..12]]; // G. C. Greubel, Sep 21 2022
-
Maple
A077385 := proc(n,k) if k < n then n^k ; else n^(2*n-k-2) ; fi ; end: for n from 1 to 10 do for k from 0 to 2*n-2 do printf("%d, ",A077385(n,k)) ; od : od : # R. J. Mathar, Jul 03 2007
-
Mathematica
Table[Join[n^Range[0,n-1],n^Range[n-2,0,-1]],{n,8}]//Flatten (* Harvey P. Dale, Oct 13 2017 *) -
SageMath
def A077385(n,k): return n^k if (k
Formula
T(n, k) = n^k for k < n, otherwise n^(2*n-k-2), for n >= 1, 0 <= k <= 2*n-2.
From G. C. Greubel, Sep 21 2022: (Start)
T(n, 0) = T(n, 2*n-2) = 1.
T(n, n-1) = A000169(n).
T(n, n) = A000272(n).
T(n, 2*n-2-k) = T(n, k).
Sum_{k=0..n-1} T(n, k) = A023037(n).
Sum_{k=0..n-2} T(n, k) = A060072(n).
Extensions
More terms from R. J. Mathar, Jul 03 2007