{"id":"n-53b59547a77aa9a8","revision":0,"first_index":1,"fingerprint":"f6-97b1acff54b6a7ed","name":"shift_x_plus_one of theta_plus_one of Triangle read by rows: 2-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","2"],["17","16","3"],["77","104","39","4"],["372","648","363","76","5"],["1915","4062","3015","960","130","6"],["10481","26108","24048","10300","2140","204","7"],["60814","173490","190302","102424","28875","4242","301","8"],["372939","1196192","1519017","983780","351365","70392","7714","424","9"],["2409837","8568680","12340713","9333496","4041240","1031058","154518","13128","576","10"],["16360786","63775820","102557073","88620220","45110840","14025060","2685459","312480","21195","760","11"],["116393205","492938998","874309755","848907444","496676455","182446242","42719061","6363456","591525","32780","979","12"],["865549453","3953161912","7657679604","8245896824","5450100700","2311896048","641869690","117188544","13960683","1060400","48917","1236","13"],["6713065156","32859105762","68955831360","81486023216","60017556170","28886507364","9294359074","2017136264","294948225","28725840","1815957","70824","1534","14"],["54190360453","282774216080","638525885091","820927472456","666412959030","358879280760","131470110772","33153073096","5772241332","690589900","55984929","2990988","99918","1876","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","36","224","1464","10088","73288","560456","4501832","37893256","333469704","3061379912","29258414024","290555146760","2992855266696"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-3","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178946,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-990926ea6d35455f"]},{"id":178947,"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-990926ea6d35455f"]},{"id":178948,"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-990926ea6d35455f"]}],"region_exact":{},"incoming_edges":[{"id":"e-990926ea6d35455f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-53b59547a77aa9a8","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":"1c15dad36a401b08bca510addcbb2fb2e5bdabb15911486e0c83baa1ff5ad49e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-62c0157c67ca5559","role":"sequence","name":"theta_plus_one of Triangle read by rows: 2-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,       2;\n   3:      17,      16,        3;\n   4:      77,     104,       39,       4;\n   5:     372,     648,      363,      76,       5;\n   6:    1915,    4062,     3015,     960,     130,       6;\n   7:   10481,   26108,    24048,   10300,    2140,     204,      7;\n   8:   60814,  173490,   190302,  102424,   28875,    4242,    301,     8;\n   9:  372939, 1196192,  1519017,  983780,  351365,   70392,   7714,   424,   9;\n  10: 2409837, 8568680, 12340713, 9333496, 4041240, 1031058, 154518, 13128, 576, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(5+4*x+x^2)*P(n-2,x) - (2+3*x+x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(5 + 4*x + x^2)*P[n - 2, x] - (2 + 3*x + x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = -(5 + 4*x + x^2)*v[i - 2] - (2 + 3*x + x^2)*deriv(v[i - 2], x) + (5 + 2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6aaec68d653cbdb0","steps":["\u03b8+1","x \u21a6 x+1"],"edges":["e-8098d1403de01053","e-990926ea6d35455f"],"text":"A143494 \u2192 \u03b8+1 \u2192 x \u21a6 x+1"},"headline":"A143494 \u2192 \u03b8+1 \u2192 x \u21a6 x+1","chain":null}
