{"id":"n-dc17eeceba263668","revision":0,"first_index":0,"fingerprint":"f6-e0b1f2627f62f8a9","name":"reversal of theta_plus_one of Riordan array (1, 3+3x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6"],["27","6"],["108","54"],["405","324","27"],["1458","1620","324"],["5103","7290","2430","108"],["17496","30618","14580","1620"],["59049","122472","76545","14580","405"],["196830","472392","367416","102060","7290"],["649539","1771470","1653372","612360","76545","1458"],["2125764","6495390","7085880","3306744","612360","30618"],["6908733","23383404","29229255","16533720","4133430","367416","5103"],["22320522","82904796","116917020","77944680","24800580","3306744","122472"],["71744535","290166786","455976378","350751060","136403190","24800580","1653372","17496"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","6","33","162","756","3402","14931","64314","273051","1145988","4764744","19656756","80561061","328316814","1331513397"],"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":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["3","-2"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177504,"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-ee410bb3de48c5cc"]},{"id":177505,"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-ee410bb3de48c5cc"]},{"id":177506,"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-ee410bb3de48c5cc"]}],"region_exact":{},"incoming_edges":[{"id":"e-ee410bb3de48c5cc","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-dc17eeceba263668","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":"d9b1f57ddcadb0fe15b771c409256ab1e3304e61e17838841bac43215165dfe8","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-5b4c394836f6a454","role":"sequence","name":"theta_plus_one of Riordan array (1, 3+3x).","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-14ea299ac4354683","kind":"fingerprint_related","node_a_id":"n-dfe0086cb1f4cb16","node_b_id":"n-dc17eeceba263668","fingerprint":"f6-e0b1f2627f62f8a9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0b1f2627f62f8a9"},"other_id":"n-dfe0086cb1f4cb16","other":{"id":"n-dfe0086cb1f4cb16","name":"derivative of reversal of Riordan array (1, 3+3x).","first_index":0,"fingerprint":"f6-e0b1f2627f62f8a9","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["3"]],"card_rows":[["3"],["18"],["81","18"],["324","162"]],"row_sums_preview":["0","0","3","18","99","486","2268","10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-016d7724d51f6416","steps":["reverse","D"],"edges":["e-450f652a77090025","e-d684e27dd6512806"],"text":"A099093 \u2192 reverse \u2192 D"},"headline":"A099093 \u2192 reverse \u2192 D","chain":null}},{"id":"l-ec7c63026e586b15","kind":"fingerprint_related","node_a_id":"n-5b4c394836f6a454","node_b_id":"n-dc17eeceba263668","fingerprint":"f6-e0b1f2627f62f8a9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0b1f2627f62f8a9"},"other_id":"n-5b4c394836f6a454","other":{"id":"n-5b4c394836f6a454","name":"theta_plus_one of Riordan array (1, 3+3x).","first_index":0,"fingerprint":"f6-e0b1f2627f62f8a9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","6"],["0","6","27"]],"card_rows":[["1"],["0","6"],["0","6","27"],["0","0","54","108"]],"row_sums_preview":["1","6","33","162","756","3402","14931","64314"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-016d7724d51f6416","steps":["\u03b8+1"],"edges":["e-3c1a49e974ad23c3"],"text":"A099093 \u2192 \u03b8+1"},"headline":"A099093 \u2192 \u03b8+1","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:      6;\n  2:     27,      6;\n  3:    108,     54;\n  4:    405,    324,     27;\n  5:   1458,   1620,    324;\n  6:   5103,   7290,   2430,    108;\n  7:  17496,  30618,  14580,   1620;\n  8:  59049, 122472,  76545,  14580,  405;\n  9: 196830, 472392, 367416, 102060, 7290;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (3*n+3+(2*n-2)*x)*P(n-1,x) - (3*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[Together[((3*n + 3 + (2*n - 2)*x)*P[n - 1, x] - (3*x + 4*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] = ((3*n + 3 + (2*n - 2)*x)*v[i - 1] - (3*x + 4*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-016d7724d51f6416","steps":["\u03b8+1","reverse"],"edges":["e-3c1a49e974ad23c3","e-ee410bb3de48c5cc"],"text":"A099093 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A099093 \u2192 \u03b8+1 \u2192 reverse","chain":null}
