{"id":"n-8437e3b60aeda119","revision":0,"first_index":0,"fingerprint":"f6-98f625797de408ed","name":"shift_x_plus_one of basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["16","9","1"],["121","90","18","1"],["1179","1057","291","30","1"],["14026","14445","4930","715","45","1"],["196783","226116","90540","16550","1485","63","1"],["3177861","3995719","1814946","394380","45080","2751","84","1"],["58019356","78730695","39662833","9882866","1354815","106106","4690","108","1"],["1181098459","1711958922","941298786","262505145","41574141","3929499","224112","7506","135","1"],["26515026561","40726792315","24149346435","7407718830","1321506165","145085787","10033065","435360","11430","165","1"],["650572403218","1052256734991","666683609680","222036060675","43813814220","5444106591","439092192","23185305","791175","16720","198","1"],["17316566815441","29342614732344","19718822378376","7058948565460","1519857206010","209771565102","19218261678","1187351352","49444560","1361635","23661","234","1"],["496889918749251","878347424077825","622365309401628","237566708166216","55223379777280","8347856859342","852285659940","59983764126","2930165667","98700030","2239666","32565","273","1"],["15288155067806104","28092043259887257","20884372140026941","8445393489829660","2101677525113166","344164226954548","38597741306124","3035142493812","169288194075","6703163467","186405219","3545542","43771","315","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","26","230","2558","34162","531538","9430822","187761470","4142596706","100275956114","2641269814966","75187037705854","2299655897593810","75197614774770002"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185507,"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-165dc26e708a30a6"]},{"id":185508,"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-165dc26e708a30a6"]},{"id":185509,"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-165dc26e708a30a6"]},{"id":185510,"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-165dc26e708a30a6"]}],"region_exact":{},"incoming_edges":[{"id":"e-165dc26e708a30a6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8437e3b60aeda119","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":"ce0b90e2a2fa709d7904bf8951ba3f28f336fa7879b05e7d395325d7192811db","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-d7b6e6dc2dbf6491","role":"sequence","name":"basis_map_A008277 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of 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:          1;\n  1:          3,          1;\n  2:         16,          9,         1;\n  3:        121,         90,        18,         1;\n  4:       1179,       1057,       291,        30,        1;\n  5:      14026,      14445,      4930,       715,       45,       1;\n  6:     196783,     226116,     90540,     16550,     1485,      63,      1;\n  7:    3177861,    3995719,   1814946,    394380,    45080,    2751,     84,    1;\n  8:   58019356,   78730695,  39662833,   9882866,  1354815,  106106,   4690,  108,   1;\n  9: 1181098459, 1711958922, 941298786, 262505145, 41574141, 3929499, 224112, 7506, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+1+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n + 1 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], 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-4f6e747b794bcc2e","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-760cba6118ea84eb","e-165dc26e708a30a6"],"text":"A028338 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A028338 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}
