{"id":"n-fb74ce37c1c8f6e0","revision":0,"first_index":0,"fingerprint":"f6-5ffbd82c13020df8","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","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":true,"rows":[["1"],["35","12"],["1225","840","144"],["42875","44100","15120","1728"],["1500625","2058000","1058400","241920","20736"],["52521875","90037500","61740000","21168000","3628800","248832"],["1838265625","3781575000","3241350000","1481760000","381024000","52254720","2985984"],["64339296875","154414312500","158826150000","90757800000","31116960000","6401203200","731566080","35831808"],["2251875390625","6176572500000","7411887000000","5082436800000","2178187200000","597445632000","102419251200","10032906240","429981696"],["78815638671875","243202542187500","333534915000000","266827932000000","137225793600000","47048843520000","10754021376000","1580182732800","135444234240","5159780352"],["2758547353515625","9457876640625000","14592152531250000","13341396600000000","8004837960000000","3293419046400000","940976870400000","184354652160000","23702740992000","1805923123200","61917364224"],["96549157373046875","364128250664062500","624219858281250000","642054711375000000","440266087800000000","211327722144000000","72455219020800000","17744135270400000","3041851760640000","347640201216000","23838185226240","743008370688"],["3379220508056640625","13903078661718750000","26217234047812500000","29962553197500000000","23113969609500000000","12679663328640000000","5071865331456000000","1490507362713600000","319394434867200000","48669628170240000","5006018897510400","312063515688960","8916100448256"],["118272717781982421875","527158399256835937500","1084440135614062500000","1363296170486250000000","1168539574702500000000","721155851816400000000","329671246544640000000","113030141672448000000","29064893572915200000","5536170204364800000","759246199455744000","70994449819238400","4056825703956480","106993205379072"],["4139545122369384765625","19869816587373046875000","44281305537574218750000","60728647594387500000000","57258439160422500000000","39262929710004000000000","20192363850859200000000","7912109917071360000000","2373632975121408000000","542544680027750400000","93007659433328640000","11595760137142272000","993922297469337600","52426670635745280","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","47","2209","103823","4879681","229345007","10779215329","506623120463","23811286661761","1119130473102767","52599132235830049","2472159215084012303","116191483108948578241","5460999706120583177327","256666986187667409334369"],"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":[["35","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192812,"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-b6c1ee270ccbd22b"]},{"id":192813,"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-73bb9b2309d35d10","e-b6c1ee270ccbd22b"]},{"id":192814,"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-73bb9b2309d35d10","e-b6c1ee270ccbd22b"]},{"id":192815,"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-b6c1ee270ccbd22b"]},{"id":192816,"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-73bb9b2309d35d10"]}],"region_exact":{},"incoming_edges":[{"id":"e-73bb9b2309d35d10","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-fb74ce37c1c8f6e0","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":"90b24ce6a55e2715d7627365511e168f1145480efb55e1f0717349aa8d13582a","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-89f02ea58ff2bc55","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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]"}]},{"id":"e-b6c1ee270ccbd22b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fb74ce37c1c8f6e0","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":"ca1980ec5fafcc2c05c22d9236bb8970330f4ed9fe336879ea0ca4200ea08a1a","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-89f02ea58ff2bc55","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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:              1;\n  1:             35,              12;\n  2:           1225,             840,             144;\n  3:          42875,           44100,           15120,            1728;\n  4:        1500625,         2058000,         1058400,          241920,           20736;\n  5:       52521875,        90037500,        61740000,        21168000,         3628800,         248832;\n  6:     1838265625,      3781575000,      3241350000,      1481760000,       381024000,       52254720,        2985984;\n  7:    64339296875,    154414312500,    158826150000,     90757800000,     31116960000,     6401203200,      731566080,      35831808;\n  8:  2251875390625,   6176572500000,   7411887000000,   5082436800000,   2178187200000,   597445632000,   102419251200,   10032906240,    429981696;\n  9: 78815638671875, 243202542187500, 333534915000000, 266827932000000, 137225793600000, 47048843520000, 10754021376000, 1580182732800, 135444234240, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (35+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(35 + 12*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (35 + 12*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-9e472291b944d56e","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-37d2e50d1bfe4331","e-73bb9b2309d35d10"],"text":"A038326 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A038326 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}
