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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"131ab436c5a0c8baab63ebdb86bf8bf057076d1cadd6832bdef02e37ac261dee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5a70a6e97131d437","role":"sequence","name":"Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:     6,     14,       8;\n  1:    18,     66,      80,      32;\n  2:    36,    168,     292,     224,       64;\n  3:   108,    648,    1548,    1840,     1088,      256;\n  4:   216,   1512,    4392,    6776,     5856,     2688,      512;\n  5:   648,   5400,   19224,   37896,    44672,    31488,    12288,     2048;\n  6:  1296,  12096,   49248,  114240,   165136,   152320,    87552,    28672,     4096;\n  7:  3888,  41472,  196128,  539712,   952368,  1117504,   871936,   436224,   126976,   16384;\n  8:  7776,  90720,  475200, 1471680,  2984160,  4139744,  3978880,  2616320,  1126400,  286720,   32768;\n  9: 23328, 303264, 1788480, 6315840, 14839200, 24355872, 28495616, 23764480, 13844480, 5365760, 1245184, 131072;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+14*x+8*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 14*x + 8*x^2, P(1,x) = 18 + 66*x + 80*x^2 + 32*x^3.","mathematica":"P[0, x_] := 6 + 14*x + 8*x^2;\nP[1, x_] := 18 + 66*x + 80*x^2 + 32*x^3;\nP[n_, x_] := P[n, x] = Expand[(6 + 14*x + 8*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 6 + 14*x + 8*x^2; v[2] = 18 + 66*x + 80*x^2 + 32*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 14*x + 8*x^2)*v[i - 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-5a70a6e97131d437","steps":["basis A013609"],"edges":["e-72b81ad6731299bb"],"text":"A064861 \u2192 basis A013609"},"headline":"A064861 \u2192 basis A013609","chain":null}