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binomial(i,j)*3^(i-j)*8^j.","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:           473,            864,             256;\n  1:          7139,          22704,           20736,             4096;\n  2:         99825,         456896,          726528,           442368,            65536;\n  3:       1332331,        7986000,        18275840,         19374080,          8847360,          1048576;\n  4:      17232457,      127903776,       383328000,        584826880,        464977920,        169869312,          16777216;\n  5:     217902003,     1930035184,      7162611456,      14310912000,      16375152640,      10415505408,        3170893824,        268435456;\n  6:    2708716769,    27891456384,    123522251776,     305604755456,     457949184000,     419203907584,      222197448704,      57982058496,       4294967296;\n  7:   33225626555,   390055214736,   2008184859648,    5929068085248,   11001771196416,   13188936499200,    10060893782016,    4570918944768,    1043677052928,      68719476736;\n  8:  403209055161,  5316100248800,  31204417178880,  107103192514560,  237162723409920,  352056678285312,   351704973312000,  229963286446080,   91418378895360,   18554258718720,   1099511627776;\n  9: 4850298400387, 70964793708336, 467816821894400, 1830659141160960, 4712540470640640, 8348127864029184, 10326995896369152, 8842867900416000, 5059192301813760, 1787737187287040, 326554953449472, 17592186044416;\n","recurrence_sentence":"The row polynomials satisfy (16*n+27+16*x)*P(n,x) = (176*n+473+(256*n+864)*x+256*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 473 + 864*x + 256*x^2.","mathematica":"P[0, x_] := 473 + 864*x + 256*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((176*n + 473 + (256*n + 864)*x + 256*x^2)*P[n - 1, x])/(16*n + 27 + 16*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 473 + 864*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((176*n + 473 + (256*n + 864)*x + 256*x^2)*v[i - 1])/(16*n + 27 + 16*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-26633b980c3508e8","steps":["basis A013609","D+1"],"edges":["e-c8a2ee9cf24a53ed","e-db20b02b130c9ea8"],"text":"A038226 \u2192 basis A013609 \u2192 D+1"},"headline":"A038226 \u2192 basis A013609 \u2192 D+1","chain":null}