A077385 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Rows n = 1..40 of the irregular triangle, flattened

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).

Sum_{k=0..2*n-2} T(n, k) = A077386(n) = 2*A060072(n) + A000169(n), n > 1. (End)

EXAMPLE

Irregular triangle begins as:

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;

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 *)

PROG

(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

(SageMath)

def A077385(n, k): return n^k if (k<n) else n^(2*n-k-2)

flatten([[A077385(n, k) for k in (0..2*n-2)] for n in (1..12)]) # G. C. Greubel, Sep 21 2022

CROSSREFS

Cf. A000169, A000272, A023037, A060072, A077386.

Sequence in context: A346249 A235453 A219206 * A337219 A220898 A336163