{"id":"n-d22ee5f40ba25a41","revision":0,"first_index":0,"fingerprint":"f6-e01d1caa8b2093c6","name":"derivative of basis_map_A013610 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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"],["18","18"],["132","216","81"],["1200","2520","1620","324"],["13152","32400","27540","9720","1215"],["169344","467712","476280","226800","51030","4374"],["2509056","7564032","8772624","5080320","1564920","244944","15309"],["42080256","136078848","174400128","116375616","44089920","9552816","1102248","52488"],["788410368","2701900800","3751557120","2792361600","1230027120","330674400","53275320","4723920","177147"],["16325591040","58768478208","87189696000","70853460480","35071898400","11041585632","2222866800","277136640","19486170","590490"],["370310676480","1390860066816","2182631463936","1907534914560","1037282241600","369173718144","87548619312","13718263680","1364031900","77944680","1948617"],["9132705054720","35602505662464","58642025435136","54511363805184","32048029728000","12594385305216","3393274876992","629405023104","79035905520","6417445320","303984252","6377292"],["243336303083520","980461377552384","1684903936794624","1651863560675328","1037666387197440","442826924956416","132305495370048","27998615893248","4179175496496","430306596720","29081160108","1160667144","20726199"],["6966451749519360","28912936389967872","51589386429235200","52991795446087680","35249147194060800","16134752892690432","5254448107708800","1237580086279680","211512242541600","25979169696480","2234284252200","127673385840","4352501790","66961566"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","36","429","5664","84027","1395540","25751205","523732320","11653107795","281770789860","7360503889725","206639458603200","6206002346169675","198576356084889300"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","12","-9"],["8","-12","0"],["-4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","6"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186791,"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-96dcb2c07f5347c1"]},{"id":186792,"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-96dcb2c07f5347c1"]},{"id":186793,"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-96dcb2c07f5347c1"]},{"id":186794,"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-96dcb2c07f5347c1"]}],"region_exact":{},"incoming_edges":[{"id":"e-96dcb2c07f5347c1","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-d22ee5f40ba25a41","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":"5458cc351d8b8cbb573ad4f96cbefd95b1bbd31e733a8fbf67e45a58d97d898a","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-dd4c9e25ea6d895a","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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:        18,         18;\n  3:       132,        216,         81;\n  4:      1200,       2520,       1620,        324;\n  5:     13152,      32400,      27540,       9720,       1215;\n  6:    169344,     467712,     476280,     226800,      51030,      4374;\n  7:   2509056,    7564032,    8772624,    5080320,    1564920,    244944,    15309;\n  8:  42080256,  136078848,  174400128,  116375616,   44089920,   9552816,  1102248,   52488;\n  9: 788410368, 2701900800, 3751557120, 2792361600, 1230027120, 330674400, 53275320, 4723920, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(12*n-12)*x+9*x^2)*P(n-2,x) + (4*n-2+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (12*n - 12)*x + 9*x^2)*P[n - 2, x] + (4*n - 2 + 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; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (12*n - 12)*x + 9*x^2)*v[i - 2] + (4*n - 2 + 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-4f6e747b794bcc2e","steps":["basis A013610","D"],"edges":["e-45a2b063eb896a2b","e-96dcb2c07f5347c1"],"text":"A028338 \u2192 basis A013610 \u2192 D"},"headline":"A028338 \u2192 basis A013610 \u2192 D","chain":null}