{"id":"n-5f1a3b5b4d2ae04f","revision":0,"first_index":1,"fingerprint":"f6-ac32c7b310f5a466","name":"one_plus_derivative 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 A154537 (S2[2,1] generalized Stirling2), 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"],["3","2"],["17","40","12"],["67","634","644","120"],["225","5968","20728","12672","1680"],["707","42866","370088","700592","297264","30240"],["2161","266616","4725444","20749696","25456176","8121984","665280"],["6531","1527530","49573740","421961640","1145380880","1004138208","253638720","17297280"],["19649","8329888","459922800","6797016320","34443568480","64500460032","43022294784","8917770240","518918400"],["59011","44046562","3935358416","93925571680","799078884640","2718039420608","3766872919296","1996669859328","348734718720","17643225600"],["177105","228376264","31862419708","1167677634816","15539464343200","86646998851840","213528641300352","229920703217664","99992338691328","15017365923840","670442572800"],["531395","1169155418","248152659668","13459553080792","266556680549600","2274017156747840","9020894005526144","16977641657021184","14718386439749376","5381485140533760","706219011671040","28158588057600"],["1594273","5936114160","1879686448488","146765953903232","4169247050522736","51814256658952704","309303775536259840","923544809190088704","1379488622250114816","989345436819222528","310015031201556480","36014915284992000","1295295050649600"],["4782915","29977105490","13953180665976","1535106626018064","60821835684874192","1061749968608751072","9088970366585329920","40307631563962039808","94380710661014015232","115207615384798588416","69827041356465862656","19045015689232650240","1979778391757107200","64764752532480000"],["14348849","150851676184","102048711615156","15555056526219904","840888505480611152","20050336826523044736","237441899294464337984","1491650061018794180608","5132399760454423671552","9721759839154424580096","9923186255521236360192","5170965743888991092736","1243347682180545146880","116705324545354137600","3497296636753920000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","69","1465","41273","1441757","59987357","2893524529","158668300593","9744944063861","662515723508917","49387067555283817","4005140040620419561","351025649060430271181","33061860745283094943629"],"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","2"]],"recurrence":{"denominator":{"coeffs":[["-3","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["6","-111","1294","-796","88"],["-3","86","-1436","552","-32"],["0","-16","528","-96","0"],["0","0","-64","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4","-106","260","-136","16"],["0","-2","84","-184","48","0"],["0","0","-16","32","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-11","118","-54","4"],["4","-86","20","0"],["0","16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-6","14","-4"],["0","2","-4","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191173,"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-0b0e07797a2585e3"]},{"id":191174,"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-0b0e07797a2585e3"]},{"id":191175,"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-0b0e07797a2585e3"]},{"id":191176,"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-0b0e07797a2585e3"]}],"region_exact":{},"incoming_edges":[{"id":"e-0b0e07797a2585e3","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-5f1a3b5b4d2ae04f","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":"7510db27d0ef5bd466d989c6b4a680b41783f57803b2756c6dba2dc4c9003681","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-945ed98594b7abe2","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 A154537 (S2[2,1] generalized Stirling2), 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:     3,        2;\n   3:    17,       40,         12;\n   4:    67,      634,        644,         120;\n   5:   225,     5968,      20728,       12672,         1680;\n   6:   707,    42866,     370088,      700592,       297264,         30240;\n   7:  2161,   266616,    4725444,    20749696,     25456176,       8121984,        665280;\n   8:  6531,  1527530,   49573740,   421961640,   1145380880,    1004138208,     253638720,      17297280;\n   9: 19649,  8329888,  459922800,  6797016320,  34443568480,   64500460032,   43022294784,    8917770240,    518918400;\n  10: 59011, 44046562, 3935358416, 93925571680, 799078884640, 2718039420608, 3766872919296, 1996669859328, 348734718720, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy (n-3+x)*P(n,x) = -(3*n-6+(16*n^2-86*n+111)*x+(64*n^3-528*n^2+1436*n-1294)*x^2+(96*n^2-552*n+796)*x^3+(32*n-88)*x^4)*P(n-2,x) - ((2*n-4)*x+(16*n^2-84*n+106)*x^2-(32*n^2-184*n+260)*x^3-(48*n-136)*x^4-16*x^5)*d/dx P(n-2,x) + (4*n-11+(16*n^2-86*n+118)*x+(20*n-54)*x^2+4*x^3)*P(n-1,x) + ((2*n-6)*x-(4*n-14)*x^2-4*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 + (16*n^2 - 86*n + 111)*x + (64*n^3 - 528*n^2 + 1436*n - 1294)*x^2 + (96*n^2 - 552*n + 796)*x^3 + (32*n - 88)*x^4)*P[n - 2, x] - ((2*n - 4)*x + (16*n^2 - 84*n + 106)*x^2 - (32*n^2 - 184*n + 260)*x^3 - (48*n - 136)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (4*n - 11 + (16*n^2 - 86*n + 118)*x + (20*n - 54)*x^2 + 4*x^3)*P[n - 1, x] + ((2*n - 6)*x - (4*n - 14)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n - 3 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(3*n - 6 + (16*n^2 - 86*n + 111)*x + (64*n^3 - 528*n^2 + 1436*n - 1294)*x^2 + (96*n^2 - 552*n + 796)*x^3 + (32*n - 88)*x^4)*v[i - 2] - ((2*n - 4)*x + (16*n^2 - 84*n + 106)*x^2 - (32*n^2 - 184*n + 260)*x^3 - (48*n - 136)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (4*n - 11 + (16*n^2 - 86*n + 118)*x + (20*n - 54)*x^2 + 4*x^3)*v[i - 1] + ((2*n - 6)*x - (4*n - 14)*x^2 - 4*x^3)*deriv(v[i - 1], x))/(n - 3 + 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-945ed98594b7abe2","steps":["D+1"],"edges":["e-0b0e07797a2585e3"],"text":"A290315 \u2192 D+1"},"headline":"A290315 \u2192 D+1","chain":null}