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1","region_exact":null,"edges":["e-2bc76bcaa23d62fb"]}],"region_exact":{},"incoming_edges":[{"id":"e-2bc76bcaa23d62fb","operation":{"index_shift":2,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-2/9\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 2","conclusion_id":"n-59f3eb88935e03d3","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds 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T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n)."},{"id":"e-07f8adddfb9eb09c","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-e92c310aea8cdd6a","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n)."}],"used_by_total":2,"links":[],"oeis":[{"oeis_id":"A051231","relation":"tail"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:                  162,                -27,                  1;\n  1:                -4374,                891,                -54,                1;\n  2:               157464,             -36450,               2835,              -90,                1;\n  3:             -7085880,            1797714,            -164025,             6885,             -135,             1;\n  4:            382637520,         -104162436,           10655064,          -535815,            14175,          -189,             1;\n  5:         -24106163760,         6944870988,         -775431468,         44411409,         -1428840,         26082,          -252,          1;\n  6:        1735643790720,      -524136874896,        62775936684,      -3973052916,        147287889,      -3306744,         44226,       -324,         1;\n  7:     -140587147048320,     44190730657296,     -5608987746300,     384593222880,     -15903371925,     415134153,      -6889050,      70470,      -405,      1;\n  8:    12652843234348800,  -4117752906204960,    548999627824296,  -40222377805500,    1815896696130,  -53265445695,    1035148653,  -13231350,    106920,   -495,    1;\n  9: -1252631480200531200, 420310380948639840, -58468716060810264, 4531015030568796, -219996150722370, 7089175819935, -155745162342, 2345052303, -23816430, 155925, -594, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n+18-x)*P(n-1,x) for n >= 1, with P(0,x) = 162 - 27*x + x^2.","mathematica":"P[0, x_] := 162 - 27*x + x^2;\nP[n_, x_] := P[n, x] = Expand[-(9*n + 18 - x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 162 - 27*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(9*n + 18 - x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 (\u22122/9)x+1 (shift +2)"],"edges":["e-2bc76bcaa23d62fb"],"text":"A028338 \u2192 x \u21a6 (\u22122/9)x+1 (shift +2)"},"headline":"A028338 \u2192 x \u21a6 (\u22122/9)x+1 (shift +2)","chain":null}