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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-96f5a9e1307ad295"]},{"id":204247,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-96f5a9e1307ad295"]},{"id":204248,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve 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binomial(n,3k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1,     8,      16;\n   2:   2,    36,     256,      896,       1536,        1024;\n   3:   4,   112,    1360,     9344,      39680,      106496,       176128,        163840,          65536;\n   4:   8,   304,    5216,    53312,     360448,     1691648,      5619712,      13205504,       21495808,        23068672,         14680064,          4194304;\n   5:  16,   768,   17024,   230912,    2140416,    14338048,     71553024,     270139392,      775028736,      1680867328,       2711617536,       3154116608,        2499805184,         1207959552,          268435456;\n   6:  32,  1856,   50432,   852480,   10037760,    87352320,    581632000,    3025960960,    12449873920,     40730624000,     105931341824,     217717932032,      349200973824,       427819008000,       386547056640,        242665652224,         94489280512,          17179869184;\n   7:  64,  4352,  140032,  2834432,   40471552,   433246208,   3607285760,   23917527040,   128233963520,    561344675840,    2016928595968,    5957614567424,    14439126401024,     28553211019264,     45612284248064,      57945551273984,      57161719742464,       42193758715904,       21921513078784,        7146825580544,         1099511627776;\n   8: 128,  9984,  371200,  8752128,  146872320,  1866289152,  18651078656,  150298705920,   993412513792,   5448956641280,   24998446104576,   96386833973248,   313040033546240,    856259712516096,   1967680210862080,    3779646382407680,    6019826162073600,     7855323385692160,     8254583545528320,     6811474534072320,      4248512929726464,      1882363906752512,       527765581332480,        70368744177664;\n   9: 256, 22528,  950272, 25575424,  493182976,  7253917696,  84587905024,  802461319168,  6305910685696,  41578949771264,  232173543096320, 1105054215962624,  4502450522292224,  15741978020937728,  47263874193293312,  121748080730046464,  268364735068504064,   503936071982120960,   800781877651701760,  1066724191035719680,   1175760560139010048,   1052962703502475264,    746542006980837376,    403072166649659392,    155655662120992768,    38280596832649216,     4503599627370496;\n  10: 512, 50176, 2367488, 71614464, 1560133632, 26069139456, 347454701568, 3792495181824, 34538924408832, 266047263604736, 1750861889929216, 9918210363097088, 48626144907362304, 207115543067492352, 768247612053651456, 2484417072092676096, 7004065626448723968, 17191931960428068864, 36647505406053056512, 67576337586345476096, 107187953717557067776, 145141525005780123648, 166071362159193882624, 158375273345971126272, 123541056128167182336, 76764981548436946944, 36533200177229463552, 12501992565580496896, 2738188573441261568, 288230376151711744;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+20*x+64*x^2+64*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 16*x^2.","mathematica":"P[1, x_] := 1 + 8*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 20*x + 64*x^2 + 64*x^3)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = (2 + 20*x + 64*x^2 + 64*x^3)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a8bf9bc375c66ce8","steps":["basis A013611"],"edges":["e-96f5a9e1307ad295"],"text":"Veronese parent of A098172 \u2192 basis A013611"},"headline":"Veronese parent of A098172 \u2192 basis A013611","chain":null}