{"id":"n-c25403f40cbbc459","revision":0,"first_index":0,"fingerprint":"f6-ff9c3889e2bd0aed","name":"shift_x_plus_one of Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","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","2"],["15","18","4"],["111","198","96","8"],["1071","2562","2004","528","16"],["12663","37998","41136","18792","3024","32"],["177255","639738","885060","577200","172320","17760","64"],["2866671","12093030","20385216","17326440","7642080","1579968","105600","128"],["52587711","253921554","505029492","530034192","311102640","98768448","14635392","631296","256"],["1078805223","5867608734","13452958224","16818013800","12356943984","5336410464","1267053312","137509632","3782400","512"],["24470577495","148040328618","384460528356","558029667312","493148585664","270391449888","89366526144","16272872448","1310781696","22682112","1024"],["608113846431","4050452914902","11752620804000","19430527617000","20082247056000","13376397355776","5696248102272","1479501475200","210167635200","12654144000","136065024","2048"],["16429992037551","119475628707906","383080760404884","710806343922000","841554672705360","660451714108416","345198040782336","117314432054784","24397855948800","2735798638080","123411876864","816328704","4096"],["479502707791383","3779942702596110","13272609960246384","27314926182200808","36459622829633040","32966521563634848","20456730717143040","8624229381679104","2385061662233088","402629467115520","35929416425472","1212894074880","4897837056","8192"],["15032820820737735","127694736183869658","487339807860099972","1101732074410564656","1637138892598003296","1676574945890177760","1205690415471183168","608205817910771712","210973906649304576","48172438339439616","6668363657226240","476151838175232","11986903154688","29386727424","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","37","413","6181","113645","2469397","61999133","1766710981","56319086285","1985514000757","76699067017853","3221569467519781","146178924382618925","7125712387919452117"],"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","-2"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-7","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185171,"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-01c376239159bcc9"]},{"id":185172,"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-01c376239159bcc9"]},{"id":185173,"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-01c376239159bcc9"]},{"id":185174,"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-01c376239159bcc9"]}],"region_exact":{},"incoming_edges":[{"id":"e-01c376239159bcc9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c25403f40cbbc459","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":"e7562ae2f33018e7a87942e35b7de67144010ff7c364ab9d036404abba2cd448","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-633aaaf6fe98dc51","role":"sequence","name":"Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","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,          2;\n  2:         15,         18,           4;\n  3:        111,        198,          96,           8;\n  4:       1071,       2562,        2004,         528,          16;\n  5:      12663,      37998,       41136,       18792,        3024,         32;\n  6:     177255,     639738,      885060,      577200,      172320,      17760,         64;\n  7:    2866671,   12093030,    20385216,    17326440,     7642080,    1579968,     105600,       128;\n  8:   52587711,  253921554,   505029492,   530034192,   311102640,   98768448,   14635392,    631296,     256;\n  9: 1078805223, 5867608734, 13452958224, 16818013800, 12356943984, 5336410464, 1267053312, 137509632, 3782400, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-1+(4*n-2)*x)*P(n-1,x) - (3+7*x+4*x^2)*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[(4*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (3 + 7*x + 4*x^2)*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] = (4*n - 1 + (4*n - 2)*x)*v[i - 1] - (3 + 7*x + 4*x^2)*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-633aaaf6fe98dc51","steps":["x \u21a6 x+1"],"edges":["e-01c376239159bcc9"],"text":"A142963 \u2192 x \u21a6 x+1"},"headline":"A142963 \u2192 x \u21a6 x+1","chain":null}
