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A236960 Given g.f. A(x) of this sequence, triangle A236961 transforms the diagonals in the table of successive iterations of A(x) such that A236961(n,0) = n^n.

%I A236960 #10 Apr 18 2023 08:49:47
%S A236960 1,1,2,5,16,79,720,10735,211802,4968491,132655760,3943593218,
%T A236960 128724395888,4567299614131,174792721389278,7170679832812100,
%U A236960 313729852611817418,14576333351368836005,716547887877448952206,37150482490370675725494,2025776434511141860123174,115890536127998971200900825
%N A236960 Given g.f. A(x) of this sequence, triangle A236961 transforms the diagonals in the table of successive iterations of A(x) such that A236961(n,0) = n^n.
%H A236960 Paul D. Hanna, <a href="/A236960/b236960.txt">Table of n, a(n) for n = 1..63</a>
%e A236960 G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 79*x^6 + 720*x^7 + 10735*x^8 + 211802*x^9 + 4968491*x^10 + 132655760*x^11 + 3943593218*x^12 +...
%e A236960 The table of coefficients in the successive iterations of A(x) begins:
%e A236960 [1,  0,   0,    0,     0,      0,       0,        0,         0, ...];
%e A236960 [1,  1,   2,    5,    16,     79,     720,    10735,    211802, ...];
%e A236960 [1,  2,   6,   21,    84,    410,    2876,    33235,    581074, ...];
%e A236960 [1,  3,  12,   54,   266,   1463,    9740,    90999,   1308954, ...];
%e A236960 [1,  4,  20,  110,   648,   4102,   28932,   248808,   2972926, ...];
%e A236960 [1,  5,  30,  195,  1340,   9705,   75264,   655599,   7059436, ...];
%e A236960 [1,  6,  42,  315,  2476,  20284,  174304,  1610487,  16952240, ...];
%e A236960 [1,  7,  56,  476,  4214,  38605,  366660,  3656975,  39586868, ...];
%e A236960 [1,  8,  72,  684,  6736,  68308,  712984,  7710392,  88021908, ...];
%e A236960 [1,  9,  90,  945, 10248, 114027, 1299696, 15223599, 185218134, ...];
%e A236960 [1, 10, 110, 1265, 14980, 181510, 2245428, 28396003, 369356822, ...]; ...
%e A236960 Then the triangle T=A236961 transforms the adjacent diagonals in the above table into each other, as illustrated by:
%e A236960 T*[1, 1,  6,  54,  648,  9705, 174304, 3656975,  88021908, ...]
%e A236960 = [1, 2, 12, 110, 1340, 20284, 366660, 7710392, 185218134, ...];
%e A236960 T*[1, 2, 12, 110, 1340, 20284, 366660,  7710392, 185218134, ...]
%e A236960 = [1, 3, 20, 195, 2476, 38605, 712984, 15223599, 369356822, ...];
%e A236960 T*[1, 3, 20, 195, 2476, 38605,  712984, 15223599, 369356822, ...]
%e A236960 = [1, 4, 30, 315, 4214, 68308, 1299696, 28396003, 701068918, ...]; ...
%e A236960 Triangle T=A236961 begins:
%e A236960 1;
%e A236960 1, 1;
%e A236960 4, 2, 1;
%e A236960 27, 11, 3, 1;
%e A236960 256, 94, 21, 4, 1;
%e A236960 3125, 1076, 217, 34, 5, 1;
%e A236960 46656, 15362, 2910, 412, 50, 6, 1;
%e A236960 823543, 262171, 47598, 6333, 695, 69, 7, 1;
%e A236960 16777216, 5198778, 915221, 116768, 12045, 1082, 91, 8, 1;
%e A236960 387420489, 117368024, 20182962, 2498414, 247151, 20871, 1589, 116, 9, 1;
%e A236960 10000000000, 2970653234, 501463686, 60678776, 5824330, 471666, 33761, 2232, 144, 10, 1; ...
%e A236960 such that column 0 equals A236961(n,0) = n^n.
%o A236960 (PARI) /* From Root Series G, Calculate T(n,k) of Triangle: */
%o A236960 {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x;
%o A236960 for(i=1, r+c-2, F=subst(F, x, G +x*O(x^(m+2)))); polcoeff(F, c));
%o A236960 N=matrix(m+1, m+1, r, c, M[r, c]);
%o A236960 P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%o A236960 /* Calculates Root Series G and then Prints ROWS of Triangle: */
%o A236960 {ROWS=12;V=[1,1];print("");print1("Root Sequence: [1, 1, ");
%o A236960 for(i=2,ROWS,V=concat(V,0);G=x*truncate(Ser(V));
%o A236960 for(n=0,#V-1,if(n==#V-1,V[#V]=n^n-T(n,0));for(k=0,n, T(n,k)));print1(V[#V]", "););
%o A236960 print1("...]");print("");print("");print("Triangle begins:");
%o A236960 for(n=0,#V-2,for(k=0,n,print1(T(n,k),", "));print(""))}
%Y A236960 Cf. A236961.
%K A236960 nonn
%O A236960 1,3
%A A236960 _Paul D. Hanna_, Feb 01 2014