{"id":"n-af2f6bf610715ee0","revision":0,"first_index":0,"fingerprint":"f6-91ad29f9b5defc65","name":"derivative of basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n.","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":[[],["3"],["39","18"],["333","378","81"],["2391","4698","2349","324"],["15657","45450","38475","11988","1215"],["96939","379458","471501","243972","54675","4374"],["578253","2877282","4827843","3627828","1330425","231822","15309"],["3359631","20397546","43735221","44152452","23285475","6565374","933849","52488"],["19141137","137637306","362604843","467041140","328870125","132256638","30174039","3621672","177147"],["107422899","894490290","2812038525","4453123716","3986212095","2127902886","687756825","131482440","13640319","590490"],["595705173","5644443186","20702795427","39200028084","43065401625","29107049814","12423115719","3345637608","549687141","50191650","1948617"],["3271488471","34784967834","146224383381","323928295236","425295009675","351856129230","189406576233","67056201864","15446686959","2223194850","181221381","6377292"],["17822273817","210253504842","998475078987","2543206501044","3908142263925","3858019887150","2534986003095","1127924552232","340236145521","68381694450","8751238947","644106492","20726199"],["96435563259","1250487429282","6629754846141","19141960404420","33855801236775","39087952102710","30585402316665","16565684404680","6260246194131","1642049354250","292442593509","33678479052","2259155691","66961566"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","57","792","9762","112785","1250919","13488762","142482036","1481524047","15214660485","154686004044","1559680532406","15616863976701","155444221042131"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"],["39","18"],["333","378","81"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-100","60","-9"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["140","18","-18"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-69","-36","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["14","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184957,"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-c72b0549273a5bb4"]},{"id":184958,"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-c72b0549273a5bb4"]},{"id":184959,"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-c72b0549273a5bb4"]}],"region_exact":{},"incoming_edges":[{"id":"e-c72b0549273a5bb4","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-af2f6bf610715ee0","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":"c5d060754b0c28c6a6eaf682ad987c16189efb7aa2491363b85a427da507405c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f5630447d9b46bac","role":"sequence","name":"basis_map_A008288 of Triangle of coefficients in expansion of (2+3x)^n.","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:        0;\n  1:        3;\n  2:       39,        18;\n  3:      333,       378,        81;\n  4:     2391,      4698,      2349,       324;\n  5:    15657,     45450,     38475,     11988,      1215;\n  6:    96939,    379458,    471501,    243972,     54675,      4374;\n  7:   578253,   2877282,   4827843,   3627828,   1330425,    231822,    15309;\n  8:  3359631,  20397546,  43735221,  44152452,  23285475,   6565374,   933849,   52488;\n  9: 19141137, 137637306, 362604843, 467041140, 328870125, 132256638, 30174039, 3621672, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(100-60*x+9*x^2)*P(n-4,x) + (140+18*x-18*x^2)*P(n-3,x) - (69+36*x+9*x^2)*P(n-2,x) + (14+6*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 39 + 18*x, P(3,x) = 333 + 378*x + 81*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[2, x_] := 39 + 18*x;\nP[3, x_] := 333 + 378*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[-(100 - 60*x + 9*x^2)*P[n - 4, x] + (140 + 18*x - 18*x^2)*P[n - 3, x] - (69 + 36*x + 9*x^2)*P[n - 2, x] + (14 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 39 + 18*x; v[4] = 333 + 378*x + 81*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(100 - 60*x + 9*x^2)*v[i - 4] + (140 + 18*x - 18*x^2)*v[i - 3] - (69 + 36*x + 9*x^2)*v[i - 2] + (14 + 6*x)*v[i - 1]); 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-5733d79f45a14084","steps":["basis A008288","D"],"edges":["e-28556ceb40718bbf","e-c72b0549273a5bb4"],"text":"A013620 \u2192 basis A008288 \u2192 D"},"headline":"A013620 \u2192 basis A008288 \u2192 D","chain":null}