{"id":"n-c45ee837fbbe8408","revision":0,"first_index":1,"fingerprint":"f6-f2c232606f089033","name":"reversal of basis_map_A013609 of Riordan array (1, 3+x).","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"],["6","3"],["36","38","10"],["216","348","186","33"],["1296","2808","2272","814","109"],["7776","21168","22968","12420","3348","360"],["46656","152928","208224","150776","61248","13236","1189"],["279936","1073088","1758240","1596432","867624","282276","50910","3927"],["1679616","7371648","14121216","15422688","10504816","4569800","1240008","191906","12970"],["10077696","49828608","109268352","139489344","114248064","62264928","22581624","5255556","712302","42837"],["60466176","332563968","821518848","1200355200","1148923008","752766080","341928944","106328024","21664300","2611774","141481"],["362797056","2196937728","6036539904","9935053056","10882850688","8331350976","4548649824","1771183152","482062608","87343704","9482184","467280"],["2176782336","14390949888","43535646720","79695539712","98324492544","86134859136","54940407040","25709672352","8760510096","2119906672","345812064","34144968","1543321"],["13060694016","93601640448","309143402496","623050195968","854940220416","843483386880","615592572288","336507905856","137783093760","41736278688","9091342416","1348614072","122111574","5097243"],["78364164096","605145489408","2166624018432","4767193626624","7201718636544","7902065507328","6494599805184","4061840825216","1942582031872","706997645952","192757214000","38177242480","5192613660","434149790","16835050"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","84","783","7299","68040","634257","5912433","55114668","513769311","4789267803","44644718160","416170266849","3879466556121","36163709805636"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-6"],["6"]]},"terms":[{"coeff":{"coeffs":[["-36","-62","-26"],["36","40","13"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-36","-44","-13"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184782,"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-b1552aacebbf5e6f"]},{"id":184783,"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-b1552aacebbf5e6f"]},{"id":184784,"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-b1552aacebbf5e6f"]}],"region_exact":{},"incoming_edges":[{"id":"e-b1552aacebbf5e6f","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-c45ee837fbbe8408","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":"b27ba4e41c67a7e064558bde5b78e764b0b3e8430175f1daed228c6b5a746e13","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-6321b10e19f62245","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+x).","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-90d0a13f9a744089","kind":"fingerprint_related","node_a_id":"n-6321b10e19f62245","node_b_id":"n-c45ee837fbbe8408","fingerprint":"f6-f2c232606f089033","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f2c232606f089033"},"other_id":"n-6321b10e19f62245","other":{"id":"n-6321b10e19f62245","name":"basis_map_A013609 of Riordan array (1, 3+x).","first_index":1,"fingerprint":"f6-f2c232606f089033","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","6"],["10","38","36"]],"card_rows":[["1"],["3","6"],["10","38","36"],["33","186","348","216"]],"row_sums_preview":["1","9","84","783","7299","68040","634257","5912433"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5d39164c4108e043","steps":["basis A013609"],"edges":["e-925e2f91c3c9b55a"],"text":"A099097 \u2192 basis A013609"},"headline":"A099097 \u2192 basis A013609","chain":null}},{"id":"l-ec4feab19eaf6060","kind":"fingerprint_related","node_a_id":"n-c45ee837fbbe8408","node_b_id":"n-1131397985fc0853","fingerprint":"f6-f2c232606f089033","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f2c232606f089033"},"other_id":"n-1131397985fc0853","other":{"id":"n-1131397985fc0853","name":"basis_map_A013610 of basis_map_A013609 of Riordan array (1, 3+3x).","first_index":0,"fingerprint":"f6-f2c232606f089033","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","18"],["90","342","324"]],"card_rows":[["1"],["9","18"],["90","342","324"],["891","5022","9396","5832"]],"row_sums_preview":["1","27","756","21141","591219","16533720","462373353","12930490971"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-016d7724d51f6416","steps":["basis A013609","basis A013610"],"edges":["e-2339e629d7f9e0b9","e-f8acb77038db5d1e"],"text":"A099093 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A099093 \u2192 basis A013609 \u2192 basis A013610","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:        6,        3;\n   3:       36,       38,        10;\n   4:      216,      348,       186,        33;\n   5:     1296,     2808,      2272,       814,       109;\n   6:     7776,    21168,     22968,     12420,      3348,      360;\n   7:    46656,   152928,    208224,    150776,     61248,    13236,     1189;\n   8:   279936,  1073088,   1758240,   1596432,    867624,   282276,    50910,    3927;\n   9:  1679616,  7371648,  14121216,  15422688,  10504816,  4569800,  1240008,  191906,  12970;\n  10: 10077696, 49828608, 109268352, 139489344, 114248064, 62264928, 22581624, 5255556, 712302, 42837;\n","recurrence_sentence":"The row polynomials satisfy (6*n-6)*P(n,x) = (36*n-36+(40*n-62)*x+(13*n-26)*x^2)*P(n-1,x) - (36*x+44*x^2+13*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((36*n - 36 + (40*n - 62)*x + (13*n - 26)*x^2)*P[n - 1, x] - (36*x + 44*x^2 + 13*x^3)*D[P[n - 1, x], x])/(6*n - 6)]];\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] = ((36*n - 36 + (40*n - 62)*x + (13*n - 26)*x^2)*v[i - 1] - (36*x + 44*x^2 + 13*x^3)*deriv(v[i - 1], x))/(6*n - 6)); 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-5d39164c4108e043","steps":["basis A013609","reverse"],"edges":["e-925e2f91c3c9b55a","e-b1552aacebbf5e6f"],"text":"A099097 \u2192 basis A013609 \u2192 reverse"},"headline":"A099097 \u2192 basis A013609 \u2192 reverse","chain":null}
