{"id":"n-62e2b222477e8804","revision":0,"first_index":0,"fingerprint":"f6-54f6f3d3ffb8a8f5","name":"shift_x_plus_one of basis_map_A013611 of Riordan array (1-x,x/(1-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["25","40","16"],["150","340","256","64"],["900","2640","2896","1408","256"],["5400","19440","27936","20032","7168","1024"],["32400","138240","245376","231936","123136","34816","4096"],["194400","959040","2025216","2373120","1666560","701440","163840","16384"],["1166400","6531840","15987456","22339584","19491840","10874880","3788800","753664","65536"],["6998400","43856640","122052096","197987328","206309376","143216640","66232320","19677184","3407872","262144"],["41990400","291133440","907739136","1676132352","2029805568","1684537344","970260480","382992384","99155968","15204352","1048576"],["251942400","1914762240","6610968576","13687750656","18883362816","18226446336","12559712256","6178996224","2126905344","487849984","67108864","4194304"],["1511654400","12496343040","47324860416","108570378240","168051179520","184892129280","148264058880","87312826368","37477416960","11434721280","2354053120","293601280","16777216"],["9069926400","81024675840","333934534656","840721711104","1442588590080","1781557493760","1629152870400","1116933193728","574115807232","218517995520","59863203840","11177820160","1275068416","67108864"],["54419558400","522427760640","2327705911296","6380068405248","12018418384896","16459699322880","16901147197440","13218210643968","7912427618304","3607571202048","1233251205120","306519736320","52361691136","5502926848","268435456"],["326517350400","3352244797440","16055946510336","47591234076672","97630783930368","146831869476864","167245680476160","146913852653568","100347408285696","53295137685504","21829792038912","6772123238400","1540249092096","242464325632","23622320128","1073741824"],["1959104102400","21419538186240","109744658251776","349771190501376","776149639888896","1271514352582656","1590801560764416","1550465837826048","1189739860328448","721160459255808","344159302975488","127951907586048","36329987506176","7615782322176","1111591223296","100931731456","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["81","810","8100","81000","810000","8100000","81000000","810000000","8100000000","81000000000","810000000000","8100000000000","81000000000000","810000000000000","8100000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["25","40","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184390,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-71df10866f9cec69"]},{"id":184391,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-5a497cb7486a225b","e-71df10866f9cec69"]},{"id":184392,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-5a497cb7486a225b","e-71df10866f9cec69"]},{"id":184393,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-71df10866f9cec69"]},{"id":184394,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-5a497cb7486a225b"]}],"region_exact":{},"incoming_edges":[{"id":"e-5a497cb7486a225b","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-62e2b222477e8804","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"6c2053f15960afce539770aa3ac70001f002b542cdaf3aaeaddf46749e5ae65a","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-84c761192e0db3c0","role":"sequence","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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]"}]},{"id":"e-71df10866f9cec69","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-62e2b222477e8804","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"825b465b5476dc8efdf4899addcbe9044e4ff189bd78c081c0b152890a722a81","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-84c761192e0db3c0","role":"sequence","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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:        25,         40,         16;\n  1:       150,        340,        256,          64;\n  2:       900,       2640,       2896,        1408,         256;\n  3:      5400,      19440,      27936,       20032,        7168,        1024;\n  4:     32400,     138240,     245376,      231936,      123136,       34816,        4096;\n  5:    194400,     959040,    2025216,     2373120,     1666560,      701440,      163840,      16384;\n  6:   1166400,    6531840,   15987456,    22339584,    19491840,    10874880,     3788800,     753664,      65536;\n  7:   6998400,   43856640,  122052096,   197987328,   206309376,   143216640,    66232320,   19677184,    3407872,    262144;\n  8:  41990400,  291133440,  907739136,  1676132352,  2029805568,  1684537344,   970260480,  382992384,   99155968,  15204352,  1048576;\n  9: 251942400, 1914762240, 6610968576, 13687750656, 18883362816, 18226446336, 12559712256, 6178996224, 2126905344, 487849984, 67108864, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 40*x + 16*x^2.","mathematica":"P[0, x_] := 25 + 40*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(6 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 25 + 40*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 4*x)*v[i - 1]); 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-7717563017cb9e5f","steps":["basis A013611","basis A007318"],"edges":["e-d3bf3843e93b142d","e-5a497cb7486a225b"],"text":"A159854 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A159854 \u2192 basis A013611 \u2192 basis A007318","chain":null}
