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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"2d6b63419319b2447a5f98035846ca03390cd736c600d616a4a5ca2f9d7db7cf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-8f767a3615460a6b","role":"sequence","name":"Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-8f767a3615460a6b","role":"sequence","name":"Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"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:      3,       1;\n  1:      7,       6,        1;\n  2:     20,      28,       11,        1;\n  3:     67,     132,       86,       20,        1;\n  4:    255,     653,      607,      243,       37,       1;\n  5:   1080,    3420,     4172,     2428,      667,      70,       1;\n  6:   5017,   18989,    28849,    22293,     9102,    1818,     135,      1;\n  7:  25287,  111608,   203772,   197720,   108642,   33080,    4972,    264,     1;\n  8: 137122,  692650,  1481015,  1737419,  1209644,  503158,  118645,  13717,   521,    1;\n  9: 794545, 4526398, 11114639, 15340444, 12997273, 6917650, 2264957, 424160, 38273, 1034, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-2,x) - (1+2*x+x^2)*d/dx P(n-2,x) + (n+1+(n+1)*x)*P(n-1,x) - (1+2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + x, P(1,x) = 7 + 6*x + x^2.","mathematica":"P[0, x_] := 3 + x;\nP[1, x_] := 7 + 6*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(n + (n - 1)*x)*P[n - 2, x] - (1 + 2*x + x^2)*D[P[n - 2, x], x] + (n + 1 + (n + 1)*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; v[2] = 7 + 6*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (n + (n - 1)*x)*v[i - 2] - (1 + 2*x + x^2)*deriv(v[i - 2], x) + (n + 1 + (n + 1)*x)*v[i - 1] - (1 + 2*x + x^2)*deriv(v[i - 1], x)); 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-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-236d0fb15dc6eca0","e-7a81e5633c7cc19a"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}