{"id":"n-7cfc47df0195bedb","revision":0,"first_index":1,"fingerprint":"f6-0833365c0b4ddd0e","name":"mobius_x_over_x_plus_one of Triangle read by rows, k!*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"],["3","7"],["9","58","81"],["27","397","1385","1399"],["81","2644","17366","40948","32289"],["243","17779","196366","811254","1441231","931687"],["729","121294","2132527","13759908","41329447","58891822","32259441"],["2187","836665","22816335","216474549","978456881","2305486971","2742082101","1303134679"],["6561","5810728","243327228","3275407320","20925145574","72517350552","140467243452","143355570664","60160827969"],["19683","40511071","2600640140","48562381804","421927539482","2018161118098","5661118184156","9305566719580","8316131752859","3124574220487"],["59049","283006690","27916936157","712839817816","8210924074946","52103334639820","198536084053970","467319494446552","667003944507221","530150154527266","180312309395601"],["177147","1979098213","301138433317","10420968360147","156359392963374","1281621528042354","6373139949134922","20145678220355142","40825277584799847","51481436663801049","36842627781440257","11445969681199159"],["531441","13847133052","3263029722226","152241397912044","2938725530745135","30530306492521848","192795947765668092","786058089704698680","2120805993859289439","3771805704340067340","4259608335696609202","2772195694255085020","792626097462398049"],["1594323","96908142283","35491709067642","2226930915694242","54813535716819265","711806747393551281","5597487979063264956","28634777998000146444","98774502149625105189","232302428543335598205","368008348822241445418","376278919245859506738","224540005776649290087","59462922484586318887"],["4782969","678285251446","387210665239131","32649334663333980","1018291529980365361","16357841323946597898","157862702074000435947","993254661539525411592","4260404590331341555395","12722824978916961665322","26508431911601520632329","37852057652836766080476","35355605897265096190611","19479458600606775149590","4804064349575887075761"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","148","3208","93328","3398560","148495168","7569290368","440950690048","28898683087360","2104377285465088","168562834887804928","14729520413382381568","1394368275802005544960","142151374514825419767808"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["3","21","49","47","16"],["0","-4","-16","-20","-8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","20","36","28","8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191567,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0)","region_exact":null,"edges":["e-05413741bc28c28e"]},{"id":191568,"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-05413741bc28c28e"]},{"id":191569,"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-05413741bc28c28e"]}],"region_exact":{},"incoming_edges":[{"id":"e-05413741bc28c28e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-7cfc47df0195bedb","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":"38fe80ea19ea6d3d75498a41d6d633ad3dd3cfd4de99a9136c45e7f1b2efb015","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f425461cbf5769b3","role":"sequence","name":"Triangle read by rows, k!*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":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0)"}]}],"used_by":[],"used_by_total":0,"links":[],"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:     3,        7;\n   3:     9,       58,         81;\n   4:    27,      397,       1385,        1399;\n   5:    81,     2644,      17366,       40948,        32289;\n   6:   243,    17779,     196366,      811254,      1441231,        931687;\n   7:   729,   121294,    2132527,    13759908,     41329447,      58891822,      32259441;\n   8:  2187,   836665,   22816335,   216474549,    978456881,    2305486971,    2742082101,    1303134679;\n   9:  6561,  5810728,  243327228,  3275407320,  20925145574,   72517350552,  140467243452,  143355570664,   60160827969;\n  10: 19683, 40511071, 2600640140, 48562381804, 421927539482, 2018161118098, 5661118184156, 9305566719580, 8316131752859, 3124574220487;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3-(4*n-21)*x-(16*n-49)*x^2-(20*n-47)*x^3-(8*n-16)*x^4)*P(n-1,x) + (4*x+20*x^2+36*x^3+28*x^4+8*x^5)*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[((3 - (4*n - 21)*x - (16*n - 49)*x^2 - (20*n - 47)*x^3 - (8*n - 16)*x^4)*P[n - 1, x] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*D[P[n - 1, x], x])/(1 + 2*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] = ((3 - (4*n - 21)*x - (16*n - 49)*x^2 - (20*n - 47)*x^3 - (8*n - 16)*x^4)*v[i - 1] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-f425461cbf5769b3","steps":["x \u21a6 x/(x+1)"],"edges":["e-05413741bc28c28e"],"text":"A225473 \u2192 x \u21a6 x/(x+1)"},"headline":"A225473 \u2192 x \u21a6 x/(x+1)","chain":null}
