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 A152285 Triangle T, read by rows, where g.f. of row n of matrix power T^n = (n^3 + y)*y^(n-1) for n>=0. 5
 1, 1, 1, -2, 4, 1, 42, -36, 9, 1, -2448, 1584, -216, 16, 1, 297120, -165600, 18000, -800, 25, 1, -64276200, 32619600, -3132000, 114000, -2250, 36, 1, 22435915320, -10679785200, 947268000, -30870000, 507150, -5292, 49, 1, -11785471407360 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..36. FORMULA T(n,k) is divisible by C(n,k)^2 for n>=k>=0. For matrix powers: [T^m](n,k)/C(n,k)^2 is an integer for all integer m and n>=k>=0. EXAMPLE Triangle T begins: 1; 1,1; -2,4,1; 42,-36,9,1; -2448,1584,-216,16,1; 297120,-165600,18000,-800,25,1; -64276200,32619600,-3132000,114000,-2250,36,1; 22435915320,-10679785200,947268000,-30870000,507150,-5292,49,1; -11785471407360,5358907814400,-449832700800,13651948800,-202507200,1778112,-10976,64,1; 8847028761338880,-3890309297817600,313670692339200,-9056483251200,125763321600,-1002855168,5249664,-20736,81,1; ... Illustrate: g.f. of row n of T^n = (n^3 + y)*y^(n-1) as follows. Matrix cube T^3 begins: 1; 3,1; 6,12,1; 0,0,27,1; <-- row 3 of T^3, g.f.: (3^3 + y)*y^2 ... Matrix power T^4 begins: 1; 4,1; 16,16,1; -12,72,36,1; 0,0,0,64,1; <-- row 4 of T^4, g.f.: (4^3 + y)*y^3 ... Matrix power T^5 begins: 1; 5,1; 30,20,1; 30,180,45,1; 240,-720,360,80,1; 0,0,0,0,125,1; <-- row 5 of T^5, g.f.: (5^3 + y)*y^4 ... Matrix power T^6 begins: 1; 6,1; 48,24,1; 162,324,54,1; -288,-576,864,96,1; -8880,32400,-9000,1200,150,1; 0,0,0,0,0,216,1; <-- row 6 of T^6, g.f.: (6^3 + y)*y^5 ... PROG (PARI) {T(n, k)=local(M, N); M=if(n==0, Mat(1), M=matrix(n, n, r, c, if(r>=c, T(r-1, c-1)))); for(j=1, n, N=matrix(#M+1, #M+1, r, c, if(r==c, 1, if(r>c, if(r<=#M, M[r, c], if(c==#M, c^2, 0))))); for(i=2, #N-1, N[ #N, #N-i]=-(N^(#N-1))[ #N, #N-i]/(#N-1)); M=N); M[n+1, k+1]} CROSSREFS Cf. columns: A152286, A152287, A152288; A152289; variant: A132870. Sequence in context: A030043 A045497 A117258 * A009417 A009451 A321330 Adjacent sequences: A152282 A152283 A152284 * A152286 A152287 A152288 KEYWORD sign,tabl AUTHOR Paul D. Hanna, Dec 09 2008 STATUS approved

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Last modified May 31 21:32 EDT 2023. Contains 363068 sequences. (Running on oeis4.)