A236961 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of g.f. of A236960 such that column 0 equals T(n,0) = n^n.
1, 1, 1, 4, 2, 1, 27, 11, 3, 1, 256, 94, 21, 4, 1, 3125, 1076, 217, 34, 5, 1, 46656, 15362, 2910, 412, 50, 6, 1, 823543, 262171, 47598, 6333, 695, 69, 7, 1, 16777216, 5198778, 915221, 116768, 12045, 1082, 91, 8, 1, 387420489, 117368024, 20182962, 2498414, 247151, 20871, 1589, 116, 9, 1, 10000000000
Offset: 0
Examples
This triangle begins: 1; 1, 1; 4, 2, 1; 27, 11, 3, 1; 256, 94, 21, 4, 1; 3125, 1076, 217, 34, 5, 1; 46656, 15362, 2910, 412, 50, 6, 1; 823543, 262171, 47598, 6333, 695, 69, 7, 1; 16777216, 5198778, 915221, 116768, 12045, 1082, 91, 8, 1; 387420489, 117368024, 20182962, 2498414, 247151, 20871, 1589, 116, 9, 1; 10000000000, 2970653234, 501463686, 60678776, 5824330, 471666, 33761, 2232, 144, 10, 1; ... in which column 0 equals T(n,0) = n^n. ILLUSTRATION. This triangle transforms diagonals in the table of coefficients in the iterations of G(x), the g.f. of A236960, that starts as: G(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 +... The table of coefficients in the successive iterations of G(x) begins: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...]; [1, 1, 2, 5, 16, 79, 720, 10735, 211802, ...]; [1, 2, 6, 21, 84, 410, 2876, 33235, 581074, ...]; [1, 3, 12, 54, 266, 1463, 9740, 90999, 1308954, ...]; [1, 4, 20, 110, 648, 4102, 28932, 248808, 2972926, ...]; [1, 5, 30, 195, 1340, 9705, 75264, 655599, 7059436, ...]; [1, 6, 42, 315, 2476, 20284, 174304, 1610487, 16952240, ...]; [1, 7, 56, 476, 4214, 38605, 366660, 3656975, 39586868, ...]; [1, 8, 72, 684, 6736, 68308, 712984, 7710392, 88021908, ...]; [1, 9, 90, 945, 10248, 114027, 1299696, 15223599, 185218134, ...]; [1, 10, 110, 1265, 14980, 181510, 2245428, 28396003, 369356822, ...]; ... Then this triangle T transforms the adjacent diagonals in the above table into each other, as illustrated by: T*[1, 1, 6, 54, 648, 9705, 174304, 3656975, 88021908, ...] = [1, 2, 12, 110, 1340, 20284, 366660, 7710392, 185218134, ...]; T*[1, 2, 12, 110, 1340, 20284, 366660, 7710392, 185218134, ...] = [1, 3, 20, 195, 2476, 38605, 712984, 15223599, 369356822, ...]; T*[1, 3, 20, 195, 2476, 38605, 712984, 15223599, 369356822, ...] = [1, 4, 30, 315, 4214, 68308, 1299696, 28396003, 701068918, ...]; ... RELATED TRIANGLE. Compare this triangle to the triangle A088956(n,k) = (n-k+1)^(n-k-1)*C(n,k), that transforms diagonals in the table of coefficients in the iterations of x/(1-x): 1; 1, 1; 3, 2, 1; 16, 9, 3, 1; 125, 64, 18, 4, 1; 1296, 625, 160, 30, 5, 1; 16807, 7776, 1875, 320, 45, 6, 1; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1829 (rows 0..60)
Crossrefs
Programs
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PARI
/* From Root Series G, Calculate T(n,k) of Triangle: */ {T(n, k) = my(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G +x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]} /* Calculates Root Series G and then Prints ROWS of Triangle: */ {ROWS=12;V=[1,1];print("");print1("Root Sequence: [1, 1, "); for(i=2,ROWS,V=concat(V,0);G=x*truncate(Ser(V)); 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]", ");); print1("...]");print("");print("");print("Triangle begins:"); for(n=0,#V-2,for(k=0,n,print1(T(n,k),", "));print(""))}