{"id":"n-1658117d1dedf548","revision":0,"first_index":1,"fingerprint":"f6-f136838b00e1c323","name":"reversal of Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","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"],["1","6"],["1","20","60"],["1","42","420","840"],["1","72","1512","10080","15120"],["1","110","3960","55440","277200","332640"],["1","156","8580","205920","2162160","8648640","8648640"],["1","210","16380","600600","10810800","90810720","302702400","259459200"],["1","272","28560","1485120","40840800","588107520","4116752640","11762150400","8821612800"],["1","342","46512","3255840","126977760","2793510720","33522128640","201132771840","502831929600","335221286400"],["1","420","71820","6511680","341863200","10666131840","195545750400","2011327718400","10559470521600","23465490048000","14079294028800"],["1","506","106260","12113640","823727520","34596555840","899510451840","14135164243200","127216478188800","593676898214400","1187353796428800","647647525324800"],["1","600","151800","21252000","1817046000","98847302400","3459655584000","77100895872000","1060137318240000","8481098545920000","35620613892864000","64764752532480000","32382376266240000"],["1","702","210600","35521200","3729726000","255113258400","11565134380800","346954031424000","6765603612768000","82690710822720000","595373117923584000","2273242813890048000","3788738023150080000","1748648318376960000"],["1","812","285012","57002400","7210803600","605707502400","34525327636800","1341555588172800","35215834189536000","610407792618624000","6714485718804864000","43949361068540928000","153822763739893248000","236650405753681920000","101421602465863680000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","81","1303","26785","669351","19674097","664400311","25330978113","1075631907655","50322142646161","2570964805355607","142389639792952801","8495817365820681703","543205619067484203825"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-10","48"],["0","8","-56"],["0","0","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-8","72"],["0","0","0","-32"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","16"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191091,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-a6fadfbae3cc127d"]},{"id":191092,"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-a6fadfbae3cc127d"]},{"id":191093,"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-a6fadfbae3cc127d"]}],"region_exact":{},"incoming_edges":[{"id":"e-a6fadfbae3cc127d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-1658117d1dedf548","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":"6772c2a28b54e62dec565db8b80ef2542bb9f356bc7de545d743f1d8184e431a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-66d7d2fd3a2f1f0d","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","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":[{"id":"l-27175bf6d770ee28","kind":"fingerprint_related","node_a_id":"n-66d7d2fd3a2f1f0d","node_b_id":"n-1658117d1dedf548","fingerprint":"f6-f136838b00e1c323","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f136838b00e1c323"},"other_id":"n-66d7d2fd3a2f1f0d","other":{"id":"n-66d7d2fd3a2f1f0d","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","first_index":1,"fingerprint":"f6-f136838b00e1c323","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","1"],["60","20","1"]],"card_rows":[["1"],["6","1"],["60","20","1"],["840","420","42","1"]],"row_sums_preview":["1","7","81","1303","26785","669351","19674097","664400311"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A292219"],"derivation":{"root":"n-66d7d2fd3a2f1f0d","steps":[],"edges":[],"text":"A292219"},"headline":"A292219: Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].","chain":null}},{"id":"l-d92101ccfede27d1","kind":"fingerprint_related","node_a_id":"n-2fd4025bef24ce96","node_b_id":"n-1658117d1dedf548","fingerprint":"f6-f136838b00e1c323","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f136838b00e1c323"},"other_id":"n-2fd4025bef24ce96","other":{"id":"n-2fd4025bef24ce96","name":"derivative of Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].","first_index":1,"fingerprint":"f6-f136838b00e1c323","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["12","2"]],"card_rows":[["1"],["12","2"],["180","60","3"],["3360","1680","168","4"]],"row_sums_preview":["0","1","14","243","5212","133925","4016106","137718679"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-be41df8499a2681b","steps":["D"],"edges":["e-c44a88a72244775f"],"text":"A048854 \u2192 D"},"headline":"A048854 \u2192 D","chain":null}}],"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: 1,   6;\n   3: 1,  20,    60;\n   4: 1,  42,   420,     840;\n   5: 1,  72,  1512,   10080,     15120;\n   6: 1, 110,  3960,   55440,    277200,     332640;\n   7: 1, 156,  8580,  205920,   2162160,    8648640,     8648640;\n   8: 1, 210, 16380,  600600,  10810800,   90810720,   302702400,    259459200;\n   9: 1, 272, 28560, 1485120,  40840800,  588107520,  4116752640,  11762150400,   8821612800;\n  10: 1, 342, 46512, 3255840, 126977760, 2793510720, 33522128640, 201132771840, 502831929600, 335221286400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(8*n-10)*x+(16*n^2-56*n+48)*x^2)*P(n-1,x) - (8*x^2+(32*n-72)*x^3)*d/dx P(n-1,x) + 16*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (8*n - 10)*x + (16*n^2 - 56*n + 48)*x^2)*P[n - 1, x] - (8*x^2 + (32*n - 72)*x^3)*D[P[n - 1, x], x] + 16*x^4*D[P[n - 1, x], {x, 2}]];\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] = (1 + (8*n - 10)*x + (16*n^2 - 56*n + 48)*x^2)*v[i - 1] - (8*x^2 + (32*n - 72)*x^3)*deriv(v[i - 1], x) + 16*x^4*deriv(deriv(v[i - 1], x), 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-66d7d2fd3a2f1f0d","steps":["reverse"],"edges":["e-a6fadfbae3cc127d"],"text":"A292219 \u2192 reverse"},"headline":"A292219 \u2192 reverse","chain":null}
