A165675 - OEIS

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A165675

Extended triangle related to the asymptotic expansions of the E(x,m=2,n).

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.)