{"id":"n-fbdfc59ab4194e8f","revision":0,"first_index":0,"fingerprint":"f6-63c2fe03aabf7727","name":"reversal of basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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":true,"rows":[["1"],["3","2"],["27","30","8"],["486","729","354","56"],["14580","27216","18639","5574","616"],["656100","1458000","1274211","549054","116904","9856"],["41334300","106288200","112351293","62623044","19444140","3192816","216832"],["3472081200","10126903500","12519866412","8518523193","3449376036","832076604","110805552","6288128"],["374984769600","1222172582400","1726841001996","1383235562088","687717970029","217491186564","42753833964","4778923248","232660736"],["50622943896000","182242598025600","289343474059860","266171486973696","156470761809963","60996336807474","15776362167084","2611831000824","251239668768","10702393856"],["8352785742840000","32904913532400000","57947258709310500","60121529898012000","40723278969170871","18826758234591138","6018894618787404","1314428396492664","187717081392864","15835316437248","599334055936"],["1653851577082320000","7074809524185480000","13678186431114279000","15786529253330179500","12091351739062636458","6456157821383493681","2453133936237512238","663522761964303540","125234684680795560","15712437107896992","1179634344370944","40155381747712"],["387001269037262880000","1786159703248905600000","3759605577291394206000","4774624573337290044000","4076512117953741111672","2465957717589685801536","1084069808968873864491","349061907262410495162","81723214410486140700","13570250373030745368","1517316968106663264","102587472534269184","3172275158069248"],["105651346447172766240000","523225715738379413760000","1190699015299449933438000","1649356221631888448964000","1552153268948402007534456","1048245571753728406093152","522819167866753658747355","195028323107774460712398","54424133002204481966004","11223214077602118429864","1662690566611947644928","167599541067668507520","10304078591305669632","291849314542370816"],["33280174130859421365600000","176015143180989828555840000","430532115687594946891530000","645761305435786554368088000","663760039211726807963537640","494725601610955060717995216","275802068483922613551290937","116852753572824842951625000","37816604145118504654805448","9304270532678342393803584","1713408220708588061717904","229039055497182030231168","21011336109434147731200","1184164864759247788032","30936027341491306496"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","65","1625","66625","4064125","345450625","39035920625","5660208490625","1024497736803125","226413999833490625","59999709955875015625","18779909216188879890625","6854666863908941160078125","2885814749705664228392890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"]]},"terms":[{"coeff":{"coeffs":[["0","2"],["3","1"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193008,"position":0,"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-6a2274b9f70ca8f4"]},{"id":193009,"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-6a2274b9f70ca8f4"]},{"id":193010,"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-6a2274b9f70ca8f4"]}],"region_exact":{},"incoming_edges":[{"id":"e-6a2274b9f70ca8f4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-fbdfc59ab4194e8f","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"ddc3292e905820b3d73a21a9d348c23880228b7b41aa08130869d51676a48d9a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4d9510ff3052f23c","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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":[{"id":"l-5e8fe05c14633b6b","kind":"fingerprint_related","node_a_id":"n-4d9510ff3052f23c","node_b_id":"n-fbdfc59ab4194e8f","fingerprint":"f6-63c2fe03aabf7727","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63c2fe03aabf7727"},"other_id":"n-4d9510ff3052f23c","other":{"id":"n-4d9510ff3052f23c","name":"basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","first_index":0,"fingerprint":"f6-63c2fe03aabf7727","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["8","30","27"]],"card_rows":[["1"],["2","3"],["8","30","27"],["56","354","729","486"]],"row_sums_preview":["1","5","65","1625","66625","4064125","345450625","39035920625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ecd77b030b8fab94","steps":["basis A013610"],"edges":["e-201bedfe8f6594c1"],"text":"A128813 \u2192 basis A013610"},"headline":"A128813 \u2192 basis A013610","chain":null}}],"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:             27,              30,               8;\n  3:            486,             729,             354,              56;\n  4:          14580,           27216,           18639,            5574,             616;\n  5:         656100,         1458000,         1274211,          549054,          116904,           9856;\n  6:       41334300,       106288200,       112351293,        62623044,        19444140,        3192816,         216832;\n  7:     3472081200,     10126903500,     12519866412,      8518523193,      3449376036,      832076604,      110805552,       6288128;\n  8:   374984769600,   1222172582400,   1726841001996,   1383235562088,    687717970029,   217491186564,    42753833964,    4778923248,    232660736;\n  9: 50622943896000, 182242598025600, 289343474059860, 266171486973696, 156470761809963, 60996336807474, 15776362167084, 2611831000824, 251239668768, 10702393856;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (3*n^2+3*n+(n^2+n+2)*x)*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[((3*n^2 + 3*n + (n^2 + n + 2)*x)*P[n - 1, x])/(2)]];\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] = ((3*n^2 + 3*n + (n^2 + n + 2)*x)*v[i - 1])/(2)); 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-ecd77b030b8fab94","steps":["basis A013610","reverse"],"edges":["e-201bedfe8f6594c1","e-6a2274b9f70ca8f4"],"text":"A128813 \u2192 basis A013610 \u2192 reverse"},"headline":"A128813 \u2192 basis A013610 \u2192 reverse","chain":null}