{"id":"n-6c844b0e40bf4f2b","revision":0,"first_index":0,"fingerprint":"f6-2d35f0238a1794fe","name":"shift_x_plus_one of basis_map_A013610 of Riordan array (1, 2+2x).","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"],["8","6"],["72","102","36"],["640","1344","936","216"],["5696","15840","16452","7560","1296"],["50688","175488","242208","166536","57024","7776"],["451072","1868928","3217248","2944728","1511136","412128","46656"],["4014080","19365888","39942144","45646848","31212864","12768192","2892672","279936"],["35721216","196669440","472684032","647688960","553341456","301786560","102596544","19875456","1679616"],["317882368","1966694400","5397221376","8622443520","8836449408","6023763936","2731242240","794178432","134369280","10077696"],["2828828672","19428532224","59939426304","109380492288","130739121792","106943149152","60624013248","23515463808","5972714496","896914944","60466176"],["25173688320","190042079232","651064541184","1336043372544","1824622184448","1741188731904","1184643522432","574608670848","194714523648","43898443776","5925685248","362797056"],["224020135936","1843779993600","6944855408640","15829414723584","24315433638912","26517222885888","21050931571776","12256598291328","5194242713088","1562486298624","316661363712","38819284992","2176782336"],["1993550594048","17765739528192","72970292035584","182899184467968","312113195163648","382907627882496","347435070460416","236063106262656","120098899666944","45184821359616","12219004846080","2248978950144","252506750976","13060694016"],["17740565839872","170181580554240","756978296684544","2069319678197760","3883800625569792","5293970578962432","5404437120867840","4197473648625024","2492263366364160","1125737315546112","380769175627776","93516383139840","15764257677312","1632586752000","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","210","3136","46844","699720","10451896","156122624","2332043280","34834322656","520329123104","7772288240640","116096643092416","1734165038662784","25903663544572800"],"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":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["10","6"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["16","20","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-ddfbccabbc6b70d5","recurrence":{"first_index":0,"initial_polynomials":[["1"],["8","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184636,"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-76936c0cbc1c6715"]},{"id":184637,"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-1833cf117492d3f7","e-76936c0cbc1c6715"]},{"id":184638,"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-1833cf117492d3f7","e-76936c0cbc1c6715"]},{"id":184639,"position":3,"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-1833cf117492d3f7"]},{"id":184640,"position":4,"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-1833cf117492d3f7"]}],"region_exact":{},"incoming_edges":[{"id":"e-1833cf117492d3f7","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6c844b0e40bf4f2b","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":"946a3491a943e7f9a4f1697b35b957464855260e787a37048651766a6290894a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0892e4c474e4320d","role":"sequence","name":"basis_map_A013610 of Riordan array (1, 2+2x).","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-ddfbccabbc6b70d5","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-6c844b0e40bf4f2b","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"f08d007b1a7c65bd35330ccc920780bb28201239bf5d266c14ab1b54dbf416b0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-76936c0cbc1c6715","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-6c844b0e40bf4f2b","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":"4ec3019fad4bd7ae71fa4362b17d21b5113ea30eb02008db2e1c0967cd631e4f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-0892e4c474e4320d","role":"sequence","name":"basis_map_A013610 of Riordan array (1, 2+2x).","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:         8,          6;\n  2:        72,        102,         36;\n  3:       640,       1344,        936,        216;\n  4:      5696,      15840,      16452,       7560,       1296;\n  5:     50688,     175488,     242208,     166536,      57024,       7776;\n  6:    451072,    1868928,    3217248,    2944728,    1511136,     412128,      46656;\n  7:   4014080,   19365888,   39942144,   45646848,   31212864,   12768192,    2892672,    279936;\n  8:  35721216,  196669440,  472684032,  647688960,  553341456,  301786560,  102596544,  19875456,   1679616;\n  9: 317882368, 1966694400, 5397221376, 8622443520, 8836449408, 6023763936, 2731242240, 794178432, 134369280, 10077696;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(2*n-10-6*x)*P(n-1,x) + (16+20*x+6*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[Together[(-(2*n - 10 - 6*x)*P[n - 1, x] + (16 + 20*x + 6*x^2)*D[P[n - 1, x], x])/(n)]];\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] = (-(2*n - 10 - 6*x)*v[i - 1] + (16 + 20*x + 6*x^2)*deriv(v[i - 1], x))/(n)); 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-f5a001953ee28e8f","steps":["basis A013610","basis A007318"],"edges":["e-937d6a83cea8d959","e-1833cf117492d3f7"],"text":"A099040 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A099040 \u2192 basis A013610 \u2192 basis A007318","chain":null}