{"id":"n-aca4ba97e8bd9633","revision":0,"first_index":0,"fingerprint":"f6-e37acfaae6dc284d","name":"one_plus_derivative of Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["13","30","9"],["34","231","180","27"],["79","1266","2700","1026","81"],["172","5607","27180","26622","5832","243"],["361","21798","206415","451116","242271","33534","729"],["742","78015","1303776","5482269","6481134","2123577","195372","2187"],["1507","264450","7278426","52987878","119855700","85336254","18334350","1150362","6561"],["3040","864159","37301112","436844880","1698218784","2309425830","1064217528","157796424","6823440","19683"],["6109","2753574","179856261","3214643112","19893748554","46415502900","40832282898","12830639112","1362122649","40665078","59049"],["12250","8621127","829506924","21762253071","202824223884","750319224462","1133740974960","679769712918","151565674122","11824778763","243045684","177147"],["24535","26659602","3701526984","138372620490","1863615936555","10317468371940","24708160056888","25509995447076","10843264218285","1769998302090","103312130400","1455085458","531441"],["49108","81720759","16112109492","838601555922","15812875494720","125466837102117","448601722163496","735290143266348","539841058718868","167779186255665","20557445697252","908161455906","8719883928","1594323"],["98257","248943558","68819786523","4896256819428","126092393288721","1386984166189218","7078373315920467","17322879659207832","20249651062272987","10904655289755114","2540773930654233","238401158214852","8025857588547","52284228462","4782969"],["196558","754985967","289716300360","27760615591761","957186397055754","14219585492205375","100052523986973228","348979531250345625","610073566311019554","524964283796040741","212562369391674384","37905615279269163","2767458977488710","71241594649389","313590579516","14348907"],["393163","2282451522","1206048384630","153768105328590","6984851363829990","137226727945537938","1295387320744204182","6212187407112376590","15481305463119485940","19974442473451267830","12974617582296458706","4031513341055083674","559869840873818010","32205647115599670","634596913318050","1881170405514","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["52","472","5152","65656","956224","15667072","285215488","5711514880","124772279296","2952888205312","75259370911744","2055120947067904","59860854447751168","1852581727168724992","60706532109640990720"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["13","30","9"],["34","231","180","27"]],"recurrence":{"denominator":{"coeffs":[["2","6"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["-10","-7","6","-45","-54"],["-6","-27","-66","-108","-54"],["0","-18","-63","-81","0"],["0","0","-27","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-5","-1","21","63","54"],["0","-3","-12","36","81","0"],["0","0","-9","27","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["9","28","27","18"],["9","27","45","0"],["0","18","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","0","-18"],["0","3","-9","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190177,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-eb8cf800e1a686aa"]},{"id":190178,"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-eb8cf800e1a686aa"]},{"id":190179,"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-eb8cf800e1a686aa"]},{"id":190180,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-eb8cf800e1a686aa"]}],"region_exact":{},"incoming_edges":[{"id":"e-eb8cf800e1a686aa","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-aca4ba97e8bd9633","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":"4b2c7f6c21a879838c66b3e9237ebb6a7829fd1f557b02d92ef6f47a9b4840ff","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-02156bdfcfec6826","role":"sequence","name":"Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","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:    13,      30,         9;\n  1:    34,     231,       180,          27;\n  2:    79,    1266,      2700,        1026,           81;\n  3:   172,    5607,     27180,       26622,         5832,          243;\n  4:   361,   21798,    206415,      451116,       242271,        33534,           729;\n  5:   742,   78015,   1303776,     5482269,      6481134,      2123577,        195372,         2187;\n  6:  1507,  264450,   7278426,    52987878,    119855700,     85336254,      18334350,      1150362,         6561;\n  7:  3040,  864159,  37301112,   436844880,   1698218784,   2309425830,    1064217528,    157796424,      6823440,       19683;\n  8:  6109, 2753574, 179856261,  3214643112,  19893748554,  46415502900,   40832282898,  12830639112,   1362122649,    40665078,     59049;\n  9: 12250, 8621127, 829506924, 21762253071, 202824223884, 750319224462, 1133740974960, 679769712918, 151565674122, 11824778763, 243045684, 177147;\n","recurrence_sentence":"The row polynomials satisfy (3*n+2+6*x)*P(n,x) = -(6*n+10+(18*n^2+27*n+7)*x+(27*n^3+63*n^2+66*n-6)*x^2+(81*n^2+108*n+45)*x^3+(54*n+54)*x^4)*P(n-2,x) - ((3*n+5)*x+(9*n^2+12*n+1)*x^2-(27*n^2+36*n+21)*x^3-(81*n+63)*x^4-54*x^5)*d/dx P(n-2,x) + (9*n+9+(18*n^2+27*n+28)*x+(45*n+27)*x^2+18*x^3)*P(n-1,x) + ((3*n+2)*x-9*n*x^2-18*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 13 + 30*x + 9*x^2, P(1,x) = 34 + 231*x + 180*x^2 + 27*x^3.","mathematica":"P[0, x_] := 13 + 30*x + 9*x^2;\nP[1, x_] := 34 + 231*x + 180*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n + 10 + (18*n^2 + 27*n + 7)*x + (27*n^3 + 63*n^2 + 66*n - 6)*x^2 + (81*n^2 + 108*n + 45)*x^3 + (54*n + 54)*x^4)*P[n - 2, x] - ((3*n + 5)*x + (9*n^2 + 12*n + 1)*x^2 - (27*n^2 + 36*n + 21)*x^3 - (81*n + 63)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (9*n + 9 + (18*n^2 + 27*n + 28)*x + (45*n + 27)*x^2 + 18*x^3)*P[n - 1, x] + ((3*n + 2)*x - 9*n*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n + 2 + 6*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 30*x + 9*x^2; v[2] = 34 + 231*x + 180*x^2 + 27*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n + 10 + (18*n^2 + 27*n + 7)*x + (27*n^3 + 63*n^2 + 66*n - 6)*x^2 + (81*n^2 + 108*n + 45)*x^3 + (54*n + 54)*x^4)*v[i - 2] - ((3*n + 5)*x + (9*n^2 + 12*n + 1)*x^2 - (27*n^2 + 36*n + 21)*x^3 - (81*n + 63)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (9*n + 9 + (18*n^2 + 27*n + 28)*x + (45*n + 27)*x^2 + 18*x^3)*v[i - 1] + ((3*n + 2)*x - 9*n*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n + 2 + 6*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-02156bdfcfec6826","steps":["D+1"],"edges":["e-eb8cf800e1a686aa"],"text":"A156366 \u2192 D+1"},"headline":"A156366 \u2192 D+1","chain":null}