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A165675 Extended triangle related to the asymptotic expansions of the E(x,m=2,n). 8
1, 1, 1, 2, 3, 1, 6, 11, 5, 1, 24, 50, 26, 7, 1, 120, 274, 154, 47, 9, 1, 720, 1764, 1044, 342, 74, 11, 1, 5040, 13068, 8028, 2754, 638, 107, 13, 1, 40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This triangle is the same as triangle A165674 except for the extra left-hand column a(n,0) = n!. The a(n) formulas for the right-hand columns generate the coefficients of this extra left-hand column, see A080663, A165676, A165677, A165678 and A165679.

Leroy Quet discovered triangle A105954 which is the reversal of our triangle.

In square format, row k gives the (n-1)-st elementary symmetric function of {k,k+1,k+2,...,k+n}, as in the Mathematica section. [Clark Kimberling, Dec 29 2011]

LINKS

Table of n, a(n) for n=0..44.

FORMULA

a(n,m) = (n-m+1)*a(n-1,m) + a(n-1,m-1), 1 <= m <= n-1, with a(n,m=0) = n! and a(n,n) = 1.

MAPLE

nmax := 8; for n from 0 to nmax do a(n, 0) := n! od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (n-m+1)*a(n-1, m) + a(n-1, m-1) od: od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, revised Nov 27 2012

MATHEMATICA

a[n_] := SymmetricPolynomial[n - 1, t[n]]; z = 10;

t[n_] := Table[k - 1, {k, 1, n}];

t1 = Table[a[n], {n, 1, z}]  (* A000142 *)

t[n_] := Table[k, {k, 1, n}];

t2 = Table[a[n], {n, 1, z}]  (* A000254 *)

t[n_] := Table[k + 1, {k, 1, n}];

t3 = Table[a[n], {n, 1, z}]  (* A001705 *)

t[n_] := Table[k + 2, {k, 1, n}];

t4 = Table[a[n], {n, 1, z}]  (* A001711 *)

t[n_] := Table[k + 3, {k, 1, n}];

t5 = Table[a[n], {n, 1, z}]  (* A001716 *)

t[n_] := Table[k + 4, {k, 1, n}];

t6 = Table[a[n], {n, 1, z}]  (* A001721 *)

t[n_] := Table[k + 5, {k, 1, n}];

t7 = Table[a[n], {n, 1, z}]  (* A051524 *)

t[n_] := Table[k + 6, {k, 1, n}];

t8 = Table[a[n], {n, 1, z}]  (* A051545 *)

t[n_] := Table[k + 7, {k, 1, n}];

t9 = Table[a[n], {n, 1, z}]  (* A051560 *)

t[n_] := Table[k + 8, {k, 1, n}];

t10 = Table[a[n], {n, 1, z}] (* A051562 *)

t[n_] := Table[k + 9, {k, 1, n}];

t11 = Table[a[n], {n, 1, z}] (* A051564 *)

t[n_] := Table[k + 10, {k, 1, n}];

t12 = Table[a[n], {n, 1, z}] (* A203147 *)

t = {t1, t2, t3, t4, t5, t6, t7, t8, t9, t10}; TableForm[t]  (* A165675 in square format *)

m[i_, j_] := t[[i]][[j]]

Flatten[Table[m[i, n + 1 - i], {n, 1, 10},

{i, 1, n}]] (* A165675 as a sequence *)

(* Clark Kimberling, Dec 29 2011 *)

CROSSREFS

A105954 is the reversal of this triangle.

A165674, A138771 and A165680 are related triangles.

A080663 equals the third right hand column.

A000142 equals the first left hand column.

A093345 are the row sums.

Sequence in context: A103136 A155856 A086960 * A138771 A121748 A174893

Adjacent sequences:  A165672 A165673 A165674 * A165676 A165677 A165678

KEYWORD

easy,nonn,tabl

AUTHOR

Johannes W. Meijer, Oct 05 2009

STATUS

approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)