{"id":"n-679a382cbe44b97b","revision":0,"first_index":0,"fingerprint":"f6-388ff5e3b0d53753","name":"shift_x_plus_one of Stirling2 triangle with scaled diagonals (powers of 6).","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":false,"rows":[["55","20","1"],["505","327","39","1"],["5497","5224","1086","64","1"],["69823","86441","27010","2710","95","1"],["1007407","1517676","648411","97040","5685","132","1"],["16157905","28518127","15584205","3210851","277235","10605","175","1"],["284214097","573904016","382413340","102509120","12123286","675920","18172","224","1"],["5432922775","12333926385","9684921636","3236100204","498608586","37786518","1466724","29196","279","1"],["112034017735","281993025412","254597241141","102554032800","19826561910","1949511816","102081882","2910720","44595","340","1"],["2476196276617","6833110241783","6965313197491","3294285546981","775662806490","95633446854","6461282982","246847722","5381805","65395","407","1"],["58332035387017","174890387789688","198471200839770","107938359227776","30208835953791","4546706527440","383836219404","18815148000","546378327","9395320","92730","480","1"],["1457666574447247","4713561486730201","5888733669845286","3621942244425154","1180813469618497","212255081367615","21841900177812","1332004631100","49373303265","1125056647","15639910","127842","559","1"],["38485034941511935","133398769940083292","181794461613443407","124776813051318064","46594812243627011","9820559886621484","1207400983519731","89480045658432","4108289369613","118940153332","2181368189","25012624","172081","644","1"],["1069787864231083297","3954202583523915615","5833476175519233177","4419490281743443271","1863832789202399925","453421963455332235","65496964095624061","5789121110524323","321858729267075","11494520814205","266742655179","4019679573","38657255","226905","735","1"],["31213730550761076769","122476135556272784416","194341074989771324856","161052720159962913152","75800965650677807996","20998336144492158816","3512934506209188424","364613117516135488","24112055344180518","1039029058244320","29631577563016","562962586368","7091301308","58005920","293880","832","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["76","872","11872","186080","3276352","63759104","1355858176","31225762304","773059428352","20446915094528","574790732959744","17098196945371136","536171562141859840","17665831366932856832","609785691961755566080"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["55","20","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191737,"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-83dd9efd9dd998f4"]},{"id":191738,"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-83dd9efd9dd998f4"]},{"id":191739,"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-83dd9efd9dd998f4"]},{"id":191740,"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-83dd9efd9dd998f4"]}],"region_exact":{},"incoming_edges":[{"id":"e-83dd9efd9dd998f4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-679a382cbe44b97b","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":"9194dcf71ff12998b08bac409c3cab42e6d0e1265169629dd30936cbdefeef81","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-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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:            55,            20,             1;\n  1:           505,           327,            39,             1;\n  2:          5497,          5224,          1086,            64,            1;\n  3:         69823,         86441,         27010,          2710,           95,           1;\n  4:       1007407,       1517676,        648411,         97040,         5685,         132,          1;\n  5:      16157905,      28518127,      15584205,       3210851,       277235,       10605,        175,         1;\n  6:     284214097,     573904016,     382413340,     102509120,     12123286,      675920,      18172,       224,       1;\n  7:    5432922775,   12333926385,    9684921636,    3236100204,    498608586,    37786518,    1466724,     29196,     279,     1;\n  8:  112034017735,  281993025412,  254597241141,  102554032800,  19826561910,  1949511816,  102081882,   2910720,   44595,   340,   1;\n  9: 2476196276617, 6833110241783, 6965313197491, 3294285546981, 775662806490, 95633446854, 6461282982, 246847722, 5381805, 65395, 407, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) + (6+6*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 55 + 20*x + x^2.","mathematica":"P[0, x_] := 55 + 20*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(7 + x)*P[n - 1, x] + (6 + 6*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] = 55 + 20*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + x)*v[i - 1] + (6 + 6*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-adb5eb667a229c4c","steps":["x \u21a6 x+1"],"edges":["e-83dd9efd9dd998f4"],"text":"A075501 \u2192 x \u21a6 x+1"},"headline":"A075501 \u2192 x \u21a6 x+1","chain":null}