This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A173210 Triangle T such that row n of T^n = row n of (I+D)^(n^2) where D is the lower diagonal matrix: D(n+1,n)=n+1, and I is the identity matrix. 9
 1, 1, 1, 4, 4, 1, 84, 36, 9, 1, 4584, 1056, 144, 16, 1, 469440, 73200, 6000, 400, 25, 1, 76982940, 9179640, 537300, 22800, 900, 36, 1, 18391183020, 1794887640, 83163780, 2598960, 67620, 1764, 49, 1, 6011375932800, 500614248960, 19475406720 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS FORMULA [T^n](n,k) = (n-k)!*C(n,k)*C(n^2,n-k) gives the k-th term of the n-th row of the n-th matrix power of this triangle T. [From Paul D. Hanna, Feb 13 2010] EXAMPLE Triangle T begins: 1; 1,1; 4,4,1; 84,36,9,1; 4584,1056,144,16,1; 469440,73200,6000,400,25,1; 76982940,9179640,537300,22800,900,36,1; 18391183020,1794887640,83163780,2598960,67620,1764,49,1; 6011375932800,500614248960,19475406720,492038400,9619680,169344,3136,64,1; 2570927357779200,187826632116480,6361668149760,137161382400,2198871360,29502144,374976,5184,81,1; ... Generator triangle I+D, with diagonal D(n+1,n)=n+1, begins: 1; 1,1; 0,2,1; 0,0,3,1; 0,0,0,4,1; ... ILLUSTRATE: row n of T^n = row n of (I+D)^(n^2). For n=2, matrix square T^2 begins: 1; 2,1; 12,8,1; <== row 2 of T^2 240,108,18,1; 12144,3264,432,32,1; ... while (I+D)^4 begins: 1; 4,1; 12,8,1; <== row 2 of (I+D)^4 = row 2 of T^2 24,36,12,1; 24,96,72,16,1; ... For n=3, matrix cube T^3 begins: 1; 3,1; 24,12,1; 504,216,27,1; <== row 3 of T^3 24408,7200,864,48,1; ... while (I+D)^9 begins: 1; 9,1; 72,18,1; 504,216,27,1; <== row 3 of (I+D)^9 = row 3 of T^3 3024,2016,432,36,1; ... For n=4, matrix power T^4 begins: 1; 4,1; 40,16,1; 912,360,36,1; 43680,13440,1440,64,1; <== row 4 of T^4 ... while (I+D)^16 begins: 1; 16,1; 240,32,1; 3360,720,48,1; 43680,13440,1440,64,1; <== row 4 of (I+D)^16 = row 4 of T^4 ... PROG (PARI) {T(n, k)=local(M=Mat(1), N, L, C=matrix(n+1, n+1, r, c, if(r==c, 1, if(r==c+1, c)))); for(i=1, n, N=M; M=matrix(#N+1, #N+1, r, c, if(r>=c, if(r<=#N, (N^(#N))[r, c], (C^((#M)^2))[r, c]))); L=sum(i=1, #M, -(M^0-M)^i/i); M=sum(i=0, #M, (L/#N)^i/i!); ); M[n+1, k+1]} CROSSREFS Cf. columns: A173211, A173212, A173213, variant: A132870. Sequence in context: A111845 A120396 A141024 * A058888 A194678 A153015 Adjacent sequences:  A173207 A173208 A173209 * A173211 A173212 A173213 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Feb 12 2010 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 25 02:56 EDT 2019. Contains 326318 sequences. (Running on oeis4.)