{"id":"n-999d2dacd9cc0a1f","revision":0,"first_index":1,"fingerprint":"f6-59373de250012a5b","name":"shift_x_plus_one of derivative of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","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":[[],["1"],["5","4"],["31","48","18"],["233","532","396","96"],["2071","6240","6930","3360","600"],["21305","79684","117540","85440","30600","4320"],["249271","1113168","2046618","1982400","1066800","302400","35280"],["3270713","16975732","37369836","45220896","32482800","13849920","3245760","322560"],["47580151","281392320","721414530","1047039840","940854600","535973760","189030240","37739520","3265920"],["760192505","5045724964","14762649780","24985898880","26957410200","19226017440","9064137600","2724019200","473558400","36288000"],["13234467511","97411443888","320247653418","619210099200","779740605600","668157336000","394559771760","158549529600","41493513600","6386688000","439084800"],["249383390393","2015808590932","7356088626876","15995643619296","23027486169600","23043839005440","16356626637120","8235038085120","2882092752000","667808064000","92207808000","5748019200"],["5057242311031","44532548205600","178605886643730","431395729084320","698850941388600","799910361169920","663327671086080","401535736842240","176098902819840","54569458944000","11341999468800","1419760742400","80951270400"],["109820924003705","1046380353258244","4574829707868420","12151516737971520","21876941980693800","28189955693259360","26747310362572800","18920686001817600","9978568875475200","3875147692416000","1077153171494400","202951061913600","23233014604800","1220496076800"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","9","97","1257","19201","338889","6795937","152738217","3804290881","104035896969","3099430193377","99927770767977","3466727189976961","128775716073426249"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","-4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-5","-4","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182139,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-9f3803ec88469a6b"]},{"id":182140,"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-9f3803ec88469a6b","e-f6462fca2a52fc29"]},{"id":182141,"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-9f3803ec88469a6b","e-f6462fca2a52fc29"]},{"id":182142,"position":3,"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-f6462fca2a52fc29"]}],"region_exact":{},"incoming_edges":[{"id":"e-9f3803ec88469a6b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-999d2dacd9cc0a1f","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"7c5667f3df72f50b7db0e54a456b28d023edeb17f86249f12d2754d38b19aea4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-2b66f014abf34c41","role":"sequence","name":"derivative of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","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]"}]},{"id":"e-f6462fca2a52fc29","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-999d2dacd9cc0a1f","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":"9bd1c9553d338424d5095b1dcd03fabc48c3031ff439c02ae19dfc4b74cf7b54","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eec36ee0e876d0f2","role":"sequence","name":"shift_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","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:        0;\n   2:        1;\n   3:        5,         4;\n   4:       31,        48,        18;\n   5:      233,       532,       396,         96;\n   6:     2071,      6240,      6930,       3360,       600;\n   7:    21305,     79684,    117540,      85440,     30600,      4320;\n   8:   249271,   1113168,   2046618,    1982400,   1066800,    302400,     35280;\n   9:  3270713,  16975732,  37369836,   45220896,  32482800,  13849920,   3245760,   322560;\n  10: 47580151, 281392320, 721414530, 1047039840, 940854600, 535973760, 189030240, 37739520, 3265920;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+4*x+2*x^2)*P(n-2,x) - (2+5*x+4*x^2+x^3)*d/dx P(n-2,x) + (5+4*x)*P(n-1,x) + (2+3*x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(2 + 4*x + 2*x^2)*P[n - 2, x] - (2 + 5*x + 4*x^2 + x^3)*D[P[n - 2, x], x] + (5 + 4*x)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(2 + 4*x + 2*x^2)*v[i - 2] - (2 + 5*x + 4*x^2 + x^3)*deriv(v[i - 2], x) + (5 + 4*x)*v[i - 1] + (2 + 3*x + x^2)*deriv(v[i - 1], 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-2cabfd060b04f15d","steps":["D","x \u21a6 x+1"],"edges":["e-196070c9ee190f69","e-9f3803ec88469a6b"],"text":"A131689 \u2192 D \u2192 x \u21a6 x+1"},"headline":"A131689 \u2192 D \u2192 x \u21a6 x+1","chain":null}
