{"id":"n-b594c60e2bbf374c","revision":0,"first_index":1,"fingerprint":"f6-9543e1a6164dd734","name":"mobius_x_over_x_plus_one of shift_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","5"],["17","43","27"],["77","295","374","157"],["372","1915","3670","3105","979"],["1915","12396","31869","40696","25824","6517"],["10481","81776","264041","451784","432211","219279","46107"],["60814","555381","2159618","4636801","5938589","4538437","1916858","345261"],["372939","3901593","17747906","45863285","73660861","75314525","47885244","17313789","2726243"],["2409837","28409971","147992153","447196268","864074945","1107371875","941501665","512191414","161814452","22623525"],["16360786","214557921","1259218690","4356208284","9839419752","15165083009","16155237708","11747956458","5582080321","1565223755","196712171"],["116393205","1680301888","10968688092","42744605784","110510030685","199058573638","254956936926","232237043094","147458765057","62167300427","15664367630","1787356765"],["865549453","13637460780","97994911510","424720728244","1236766883507","2549509140826","3815611078896","4177859015626","3322064762666","1871104224615","708676573122","162079086457","16929897395"],["6713065156","114607946857","898815017580","4288033168134","13886250890082","32238367336319","55209465390650","70624959363388","67493595441271","47597676728174","24079461334213","8276235422820","1732387678280","166808851541"],["54190360453","996346085571","8467114333913","44086951918242","157149032330863","405710657219581","782418874355119","1145171252490836","1278424229597087","1083393945289798","686140380608827","314946859460076","99055810396895","19109918560263","1706299041211"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","87","903","10041","119217","1505679","20151759","284786385","4235586105","66098058855","1079350363191","18397819313097","326613377634465","6026831862048735"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["6","17","15","4"],["-1","-4","-5","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","5","9","7","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181000,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-2773a95702f3aa8b"]},{"id":181001,"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-2773a95702f3aa8b"]},{"id":181002,"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 real-rootedness)","region_exact":null,"edges":["e-2773a95702f3aa8b"]}],"region_exact":{},"incoming_edges":[{"id":"e-2773a95702f3aa8b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-b594c60e2bbf374c","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"7dc26fa21fa1c6d76cee6c3c911476e3548ef76e8170158bb5b7ee0af9fc72e5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c13c144ce0d4f57d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind.","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;\n   2:       4,        5;\n   3:      17,       43,        27;\n   4:      77,      295,       374,       157;\n   5:     372,     1915,      3670,      3105,       979;\n   6:    1915,    12396,     31869,     40696,     25824,       6517;\n   7:   10481,    81776,    264041,    451784,    432211,     219279,     46107;\n   8:   60814,   555381,   2159618,   4636801,   5938589,    4538437,   1916858,    345261;\n   9:  372939,  3901593,  17747906,  45863285,  73660861,   75314525,  47885244,  17313789,   2726243;\n  10: 2409837, 28409971, 147992153, 447196268, 864074945, 1107371875, 941501665, 512191414, 161814452, 22623525;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(n-6+(4*n-17)*x+(5*n-15)*x^2+(2*n-4)*x^3)*P(n-1,x) + (1+5*x+9*x^2+7*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 6 + (4*n - 17)*x + (5*n - 15)*x^2 + (2*n - 4)*x^3)*P[n - 1, x] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - 6 + (4*n - 17)*x + (5*n - 15)*x^2 + (2*n - 4)*x^3)*v[i - 1] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + x)); 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-a2b8ff03da0c14a3","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-6768e07b6c7820a7","e-2773a95702f3aa8b"],"text":"A143495 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A143495 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}