{"id":"n-c18c59cf0567612d","revision":0,"first_index":1,"fingerprint":"f6-42411ce3f1f92da1","name":"one_plus_derivative of Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.","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","1"],["17","15","1"],["101","213","63","1"],["561","2807","1884","255","1"],["2965","31929","46532","14750","1023","1"],["15241","323403","941427","618430","109443","4095","1"],["77229","3028573","16146597","20371695","7332847","790755","16383","1"],["388577","26929935","245485606","548439569","370138674","81526081","5633702","65535","1"],["1948517","231367041","3425508306","12652000212","14831873244","6038516358","872385774","39829284","262143","1"],["9755385","1942506899","44955991125","260214382872","495416949330","346135525974","91874722266","9117971520","280378053","1048575","1"],["48805597","16057068069","564368115851","4913533187853","14392355739714","16162020894810","7305514101030","1334132436114","93891859377","1968938081","4194303","1"],["244091473","131336023575","6857896897440","87002682469331","375782514693111","642514311391518","463838593264080","143636515615854","18765254241639","957603723659","13807732704","16777215","1"],["1220596629","1066554019657","81344301971568","1466858230014426","9036224272656365","22528527393814047","24613070916778056","12126256616533164","2683155921485823","258164186519415","9705105110984","96754792602","67108863","1"],["6103286249","8619094263963","947566488304807","23813528510523010","203768161074958341","715731569135986217","1132779362362945779","841664857566780396","295690772613340635","48273399870859933","3496870906143909","97945198123058","677686201847","268435455","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","33","378","5508","97200","2012040","47764080","1278607680","38093690880","1249949232000","44783895340800","1739500776921600","72804471541401600","3266273336880153600"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["-11","6"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["80","-460","1244","-900","144"],["-30","246","-954","540","-54"],["0","-36","270","-81","0"],["0","0","-27","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","48","-348","570","-324","54"],["0","-18","189","-252","81","0"],["0","0","-27","27","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-71","251","-156","18"],["21","-123","45","0"],["0","18","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-33","51","-18"],["0","9","-9","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186772,"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-a8faa01e318ec3dd"]},{"id":186773,"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-a8faa01e318ec3dd"]},{"id":186774,"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-a8faa01e318ec3dd"]},{"id":186775,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-a8faa01e318ec3dd"]}],"region_exact":{},"incoming_edges":[{"id":"e-a8faa01e318ec3dd","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-c18c59cf0567612d","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"8ae390b90eea4126873b9cd59df4e11aa3cd38b694b7737b3f9857af8f80a06d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-daa3b3e1ead00edc","role":"sequence","name":"Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.","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:       3,         1;\n   3:      17,        15,          1;\n   4:     101,       213,         63,           1;\n   5:     561,      2807,       1884,         255,           1;\n   6:    2965,     31929,      46532,       14750,        1023,          1;\n   7:   15241,    323403,     941427,      618430,      109443,       4095,         1;\n   8:   77229,   3028573,   16146597,    20371695,     7332847,     790755,     16383,        1;\n   9:  388577,  26929935,  245485606,   548439569,   370138674,   81526081,   5633702,    65535,      1;\n  10: 1948517, 231367041, 3425508306, 12652000212, 14831873244, 6038516358, 872385774, 39829284, 262143, 1;\n","recurrence_sentence":"The row polynomials satisfy (3*n-11+6*x)*P(n,x) = -(30*n-80+(36*n^2-246*n+460)*x+(27*n^3-270*n^2+954*n-1244)*x^2+(81*n^2-540*n+900)*x^3+(54*n-144)*x^4)*P(n-2,x) - ((18*n-48)*x+(27*n^2-189*n+348)*x^2-(27*n^2-252*n+570)*x^3-(81*n-324)*x^4-54*x^5)*d/dx P(n-2,x) + (21*n-71+(18*n^2-123*n+251)*x+(45*n-156)*x^2+18*x^3)*P(n-1,x) + ((9*n-33)*x-(9*n-51)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 80 + (36*n^2 - 246*n + 460)*x + (27*n^3 - 270*n^2 + 954*n - 1244)*x^2 + (81*n^2 - 540*n + 900)*x^3 + (54*n - 144)*x^4)*P[n - 2, x] - ((18*n - 48)*x + (27*n^2 - 189*n + 348)*x^2 - (27*n^2 - 252*n + 570)*x^3 - (81*n - 324)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (21*n - 71 + (18*n^2 - 123*n + 251)*x + (45*n - 156)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 33)*x - (9*n - 51)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 11 + 6*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = (-(30*n - 80 + (36*n^2 - 246*n + 460)*x + (27*n^3 - 270*n^2 + 954*n - 1244)*x^2 + (81*n^2 - 540*n + 900)*x^3 + (54*n - 144)*x^4)*v[i - 2] - ((18*n - 48)*x + (27*n^2 - 189*n + 348)*x^2 - (27*n^2 - 252*n + 570)*x^3 - (81*n - 324)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (21*n - 71 + (18*n^2 - 123*n + 251)*x + (45*n - 156)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 33)*x - (9*n - 51)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 11 + 6*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-daa3b3e1ead00edc","steps":["D+1"],"edges":["e-a8faa01e318ec3dd"],"text":"A225117 \u2192 D+1"},"headline":"A225117 \u2192 D+1","chain":null}