{"id":"n-d16ab74d3d5eb9b8","revision":0,"first_index":1,"fingerprint":"f6-b86b6dd33b0582b0","name":"reversal of Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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","3"],["1","10","9"],["1","21","79","27"],["1","36","310","580","81"],["1","55","850","3990","4141","243"],["1","78","1895","16740","48031","29230","729"],["1","105","3689","52745","299131","557571","205339","2187"],["1","136","6524","137592","1301286","5044536","6338620","1439560","6561"],["1","171","10740","313740","4465902","29768970","82006660","71164380","10083481","19683"],["1","210","16725","646680","12936882","132484716","647617090","1301264280","792891661","70604050","59049"],["1","253","24915","1232055","32983962","481780530","3694765558","13605988990","20311855861","8792412321","494287399","177147"],["1","300","35794","2203740","76105887","1504283352","16702839868","98585596824","278825646671","313470250236","97210822930","3460188940","531441"],["1","351","49894","3742882","162051747","4167989397","63335623780","549562273260","2546294373623","5611157536985","4799264576470","1072779241170","24221854021","1594323"],["1","406","67795","6087900","322995673","10488007530","209215252675","2512966610440","17384475751643","64175928130314","111411257779185","73061747888220","11824793506891","169554572470","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","20","128","1008","9280","96704","1120768","14274816","197833728","2958521344","47415508992","809838505984","14670950907904","280760761434112"],"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","-5"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183598,"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-adec82b915899af2"]},{"id":183599,"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-adec82b915899af2"]},{"id":183600,"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-adec82b915899af2"]}],"region_exact":{},"incoming_edges":[{"id":"e-adec82b915899af2","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-d16ab74d3d5eb9b8","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":"5e40895faac67752480e38c311c592e6cd15e85b20a00dc9708ec0c6b92cb637","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-7d6d6baad7692ddc","role":"sequence","name":"Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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-f16ec7788ade8f9e","kind":"fingerprint_related","node_a_id":"n-7d6d6baad7692ddc","node_b_id":"n-d16ab74d3d5eb9b8","fingerprint":"f6-b86b6dd33b0582b0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b86b6dd33b0582b0"},"other_id":"n-7d6d6baad7692ddc","other":{"id":"n-7d6d6baad7692ddc","name":"Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","first_index":1,"fingerprint":"f6-b86b6dd33b0582b0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["9","10","1"]],"card_rows":[["1"],["3","1"],["9","10","1"],["27","79","21","1"]],"row_sums_preview":["1","4","20","128","1008","9280","96704","1120768"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A225469"],"derivation":{"root":"n-7d6d6baad7692ddc","steps":[],"edges":[],"text":"A225469"},"headline":"A225469: Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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,   3;\n   3: 1,  10,     9;\n   4: 1,  21,    79,     27;\n   5: 1,  36,   310,    580,      81;\n   6: 1,  55,   850,   3990,    4141,      243;\n   7: 1,  78,  1895,  16740,   48031,    29230,      729;\n   8: 1, 105,  3689,  52745,  299131,   557571,   205339,     2187;\n   9: 1, 136,  6524, 137592, 1301286,  5044536,  6338620,  1439560,     6561;\n  10: 1, 171, 10740, 313740, 4465902, 29768970, 82006660, 71164380, 10083481, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-5)*x)*P(n-1,x) - 4*x^2*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[(1 + (4*n - 5)*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\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 + (4*n - 5)*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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-7d6d6baad7692ddc","steps":["reverse"],"edges":["e-adec82b915899af2"],"text":"A225469 \u2192 reverse"},"headline":"A225469 \u2192 reverse","chain":null}
