{"id":"n-76d87447f0c0edac","revision":0,"first_index":1,"fingerprint":"f6-6c53b1c3395f48c0","name":"mobius_x_over_x_plus_one of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 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"],["4","27","27"],["28","306","675","405"],["280","4314","15390","19845","8505"],["3640","73224","370062","759780","688905","229635"],["58240","1456776","9679068","27875988","39803400","27785835","7577955"],["1106560","33272112","276939324","1041862500","2070185040","2258230590","1280674395","295540245"],["24344320","858360240","8652648096","40605917220","104767753860","158081193270","139297968810","66496555125","13299311025"],["608608000","24691383360","293982898176","1666138572648","5338219912740","10409190131340","12629222776170","9327250132200","3843500886225","678264862275"],["17041024000","783691124160","10810234188864","72230511522312","278508200842872","671800411982700","1049357803614120","1063237949733960","675551802825900","244853615281275","38661097149675"],["528271744000","27207852013440","428191545457728","3310819594258176","15009534120180600","43435687989133440","83664952152261480","108805621370486040","94570198441058340","52707962447390250","17049543843006675","2435649120429525"],["17961239296000","1025708307281280","18188768046833280","160349287462326144","839425451269925520","2848806349236207360","6565797599363402400","10502216591598852120","11686991927699088000","8889397615754305650","4411772440138012950","1288458384707218725","168059789309637225"],["664565853952000","41728820623795200","825196774919644800","8194561785374500224","48828612910418005392","190948628175048984240","514954410823742329440","983379025476447359760","1343039503085575495200","1305201938853318105600","882288996129031040550","394604385299028204300","105037368318523265625","12604484198222791875"],["26582634158080000","1822251052689177600","39836848574374444800","441142972679493461376","2957010522395835791808","13138919646790266424368","40751427633754847209200","90883385203487828711760","147907406802493015357680","176089137106820295758400","151819163509911122124900","92330927848119293833500","37578168889634883510000","9188668980504415276875","1020963220056046141875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","58","1414","48334","2125246","114237262","7257810766","532084051966","44211070163134","4105812359289838","421445896747419694","47380507874132386654","5789949505216127474206","764148008018908565304142"],"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":[["-5","-13","-11","-3"],["3","9","9","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","9","9","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190750,"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":true,"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-68f963189369cd37"]},{"id":190751,"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-68f963189369cd37"]},{"id":190752,"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-68f963189369cd37"]}],"region_exact":{},"incoming_edges":[{"id":"e-68f963189369cd37","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-76d87447f0c0edac","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":"2b001226171a156d4ece9cb1fef73d0189cff8ed6e7b4e38fb71be3ce23df15e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d0e888e78d074358","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 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":[],"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:         4,          27,           27;\n   4:        28,         306,          675,           405;\n   5:       280,        4314,        15390,         19845,          8505;\n   6:      3640,       73224,       370062,        759780,        688905,         229635;\n   7:     58240,     1456776,      9679068,      27875988,      39803400,       27785835,        7577955;\n   8:   1106560,    33272112,    276939324,    1041862500,    2070185040,     2258230590,     1280674395,     295540245;\n   9:  24344320,   858360240,   8652648096,   40605917220,  104767753860,   158081193270,   139297968810,   66496555125,   13299311025;\n  10: 608608000, 24691383360, 293982898176, 1666138572648, 5338219912740, 10409190131340, 12629222776170, 9327250132200, 3843500886225, 678264862275;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3*n-5+(9*n-13)*x+(9*n-11)*x^2+(3*n-3)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*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*n - 5 + (9*n - 13)*x + (9*n - 11)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*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*n - 5 + (9*n - 13)*x + (9*n - 11)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*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-d0e888e78d074358","steps":["x \u21a6 x/(x+1)"],"edges":["e-68f963189369cd37"],"text":"A290595 \u2192 x \u21a6 x/(x+1)"},"headline":"A290595 \u2192 x \u21a6 x/(x+1)","chain":null}