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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"3c92710be7e47230d37ea2eb1f0e69d84ef42416055ede805c15741da7a0c691","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3b748f1f6d80b05e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:          1;\n  1:          9,          12;\n  2:         89,         228,          144;\n  3:        945,        3528,         4320,         1728;\n  4:      10577,       51576,        92736,        72576,        20736;\n  5:     123129,      739380,      1746720,      2021760,      1140480,       248832;\n  6:    1476841,    10524348,     30756240,     47027520,     39502080,     17169408,      2985984;\n  7:   18130401,   149458512,    520140096,    987880320,   1101703680,    717631488,    250822656,     35831808;\n  8:  226739489,  2121913776,   8565629184,  19434594816,  27023238144,  23479787520,  12373917696,   3583180800,   429981696;\n  9: 2878666857, 30144442140, 138442322880, 365253235200, 608480087040, 661482049536, 467031785472, 205101268992, 50307858432, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(49*n-49+(168*n-168)*x+(144*n-144)*x^2)*P(n-2,x) + (18*n+9+(24*n+12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(49*n - 49 + (168*n - 168)*x + (144*n - 144)*x^2)*P[n - 2, x] + (18*n + 9 + (24*n + 12)*x)*P[n - 1, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(49*n - 49 + (168*n - 168)*x + (144*n - 144)*x^2)*v[i - 2] + (18*n + 9 + (24*n + 12)*x)*v[i - 1])/(n + 2)); 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-8ce61d302cec295b","steps":["basis A013610","basis A013609"],"edges":["e-72b4d9a78d1bd227","e-ba3aaa374f205456"],"text":"A114687 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A114687 \u2192 basis A013610 \u2192 basis A013609","chain":null}