{"id":"n-9f4b706e6d6d188f","revision":0,"first_index":0,"fingerprint":"f6-4cecae6458dfb8cb","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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","7"],["18","72","70"],["162","882","1566","910"],["1944","13176","32904","35976","14560"],["29160","234576","743904","1164816","901944","276640"],["524880","4863888","18550944","37332576","41860944","24822288","6086080"],["11022480","115263648","511167024","1247757696","1812394224","1567791648","748364880","152152000"],["264539520","3074956992","15495390720","44258861376","78434676864","88374037824","61858923264","24605864640","4260256000"],["7142567040","91224563904","513699216192","1675346369472","3489760977984","4817574004032","4409786100672","2581984966464","877808715840","132067936000"],["214277011200","2979584196480","18512611658496","67726164434688","161654605901568","263179093372416","296091099157248","227392276416768","114121750334976","33807534418560","4490309824000"],["7071141369600","106254527898240","721160800000128","2919930057709056","7840470078835200","14666130499647744","19508632726968576","18459315790571520","12179531988474624","5338153398206592","1399058997678720","166141463488000"],["254561089305600","4108008659120640","30211969915934208","133963914077531136","399054125146429440","841599501140726784","1288955998156778496","1444880677539317760","1176835783207947264","679354801874422272","263892259844697600","61943452592716800","6645658539520000"],["9927882482918400","171158464545845760","1354911199063621632","6523707355408885248","21323559186044587008","49981707925784810496","86458062477165613056","111775454344774867968","108026464679300606976","77098775951044200960","39504054614543364096","13763161824437952000","2922749144528102400","285763317199360000"],["416971064282572800","7645338105139768320","64779559729781013504","336321624084099775488","1195680024162681375744","3080115455441013043200","5930397188627057031168","8671639956430162655232","9678782416390269419520","8207365965191684361216","5205713987558854536192","2395239308895388732416","755860907994326092800","146448519970665830400","13145112591170560000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","160","3520","98560","3351040","134041600","6165913600","320627507200","18596395417600","1190169306726400","83311851470848000","6331700711784448000","519199458366324736000","45689552336236576768000"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","4","4"],["3","6","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189167,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-106007d940a76a97"]},{"id":189168,"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-106007d940a76a97"]},{"id":189169,"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-106007d940a76a97"]}],"region_exact":{},"incoming_edges":[{"id":"e-106007d940a76a97","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-9f4b706e6d6d188f","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"42c256cbccb0d4320c7759725151189ee9127d047eabb31870bd5aae0b12c241","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7ba399ade8b53e3c","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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":[],"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,           7;\n  2:         18,          72,           70;\n  3:        162,         882,         1566,           910;\n  4:       1944,       13176,        32904,         35976,         14560;\n  5:      29160,      234576,       743904,       1164816,        901944,        276640;\n  6:     524880,     4863888,     18550944,      37332576,      41860944,      24822288,       6086080;\n  7:   11022480,   115263648,    511167024,    1247757696,    1812394224,    1567791648,     748364880,     152152000;\n  8:  264539520,  3074956992,  15495390720,   44258861376,   78434676864,   88374037824,   61858923264,   24605864640,   4260256000;\n  9: 7142567040, 91224563904, 513699216192, 1675346369472, 3489760977984, 4817574004032, 4409786100672, 2581984966464, 877808715840, 132067936000;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (3*n+(6*n+4)*x+(3*n+4)*x^2)*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 + (6*n + 4)*x + (3*n + 4)*x^2)*P[n - 1, x])/(1 + x)]];\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 + (6*n + 4)*x + (3*n + 4)*x^2)*v[i - 1])/(1 + x)); 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-8229dbdd6a4b5cd9","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-6ce7f219250b8b1f","e-106007d940a76a97"],"text":"A225470 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A225470 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}
