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 A089503 Triangle of numbers used for basis change between certain falling factorials. 2
 1, 1, 4, 1, 12, 30, 1, 24, 168, 336, 1, 40, 540, 2880, 5040, 1, 60, 1320, 13200, 59400, 95040, 1, 84, 2730, 43680, 360360, 1441440, 2162160, 1, 112, 5040, 117600, 1528800, 11007360, 40360320, 57657600, 1, 144, 8568, 274176, 5140800, 57576960 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Used to relate array A078739 ((2,2)-Stirling2) to triangle A071951 (Legendre-Stirling). LINKS Wolfdieter Lang, First 9 rows. FORMULA fallfac(x+n-1, 2*n) = Sum_{m=1..n} a(n, m)*fallfac(x, 2*n-(m-1)), n>=1 where fallfac(x, k) := Product_{j=1..k} (x+1-j), with fallfac(n, k) = A068424(n, k) (falling factorials). a(n, m) = 0 if n < m. T(n, m) = binomial(n-1, m-1)*binomial(2n, m-1)*m!, for 1 <= m <= n, with binomial(n, m) = A007318. - Stefano Negro, Nov 10 2021 EXAMPLE The triangle begins: n\m 1   2    3      4       5        6        7        8 ... 1:  1 2:  1   4 3:  1  12   30 4:  1  24  168    336 5:  1  40  540   2880    5040 6:  1  60 1320  13200   59400    95040 7:  1  84 2730  43680  360360  1441440  2162160 8:  1 112 5040 117600 1528800 11007360 40360320 57657600 ... Row 9:  1 144 8568 274176 5140800 57576960 374250240 1283143680 1764322560 Row 10: 1 180 13680 574560 14651280 234420480 2344204800 14065228800 45711993600 60949324800. Reformatted - Wolfdieter Lang, Apr 10 2013 n=3: fallfac(x+2,6) = 1*fallfac(x,6) + 12*fallfac(x,5) + 30*fallfac(x,4). MATHEMATICA eq[n_, x_] := Sum[FactorialPower[x, 1 - m + 2*n]*a[n, m], {m, 1, n}] == FactorialPower[x + n - 1, 2*n]; eq[n_] := Table[eq[n, x], {x, n + 1, 2*n}]; row[n_] := First[Table[a[n, m], {m, 1, n}] /. Solve[eq[n]]]; Array[row, 10] // Flatten (* Jean-François Alcover, Sep 02 2016 *) a[n_, m_]:= Binomial[n-1, m-1]*Binomial[2n, m-1]*Gamma[m]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] (* Stefano Negro, Nov 10 2021 *) CROSSREFS Sequence in context: A105197 A157398 A306299 * A019236 A019237 A019238 Adjacent sequences:  A089500 A089501 A089502 * A089504 A089505 A089506 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Dec 01 2003 STATUS approved

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Last modified August 9 00:15 EDT 2022. Contains 356016 sequences. (Running on oeis4.)