{"id":"n-fdfe65b0aa25bc21","revision":0,"first_index":1,"fingerprint":"f6-85a1027af8cb994c","name":"derivative of basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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":[[],["3"],["18","18"],["75","162","81"],["270","918","972","324"],["900","4212","6966","4860","1215"],["2862","17118","38880","42120","21870","4374"],["8817","64314","186705","277020","222345","91854","15309"],["26550","228726","810648","1535760","1683990","1071630","367416","52488"],["78585","781164","3274506","7572852","10588725","9185400","4837644","1417176","177147"],["229500","2586276","12528108","34280820","58480380","64603980","46294416","20785248","5314410","590490"],["663048","8354664","45948060","145382040","293405490","395155908","360159534","219662280","85916295","19486170","1948617"],["1898748","26456004","162912060","585657540","1366918740","2177123508","2417229864","1873034280","993794670","344255670","70150212","6377292"],["5397375","82405836","561840138","2263238820","6003566505","11057122080","14511462336","13687085808","9216958410","4325929740","1344545730","248714388","20726199"],["15246810","253145754","1893434616","8451419472","25130926350","52607082150","79817084424","88941808296","72850995162","43356728250","18239055120","5140097352","870500358","66961566"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","36","318","2484","18153","127224","866364","5777208","37913199","245693628","1576082106","10025808588","63328993365","397634485680"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["-12"],["6"]]},"terms":[{"coeff":{"coeffs":[["9","9"],["9","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","18","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180029,"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-ac8bcc7f7f30472d"]},{"id":180030,"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-ac8bcc7f7f30472d"]},{"id":180031,"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-ac8bcc7f7f30472d"]},{"id":180032,"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-ac8bcc7f7f30472d"]}],"region_exact":{},"incoming_edges":[{"id":"e-ac8bcc7f7f30472d","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-fdfe65b0aa25bc21","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":"4ac03873f27d757982bda2a27d614e0a6be49d9c7e863c9f703587e31679e4c3","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-57cc3445e053f3c8","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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:     3;\n   3:    18,     18;\n   4:    75,    162,      81;\n   5:   270,    918,     972,     324;\n   6:   900,   4212,    6966,    4860,     1215;\n   7:  2862,  17118,   38880,   42120,    21870,    4374;\n   8:  8817,  64314,  186705,  277020,   222345,   91854,   15309;\n   9: 26550, 228726,  810648, 1535760,  1683990, 1071630,  367416,   52488;\n  10: 78585, 781164, 3274506, 7572852, 10588725, 9185400, 4837644, 1417176, 177147;\n","recurrence_sentence":"The row polynomials satisfy (6*n-12)*P(n,x) = (9*n+9+(9*n+9)*x)*P(n-1,x) + (5+18*x+9*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n + 9 + (9*n + 9)*x)*P[n - 1, x] + (5 + 18*x + 9*x^2)*D[P[n - 1, x], x])/(6*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = ((9*n + 9 + (9*n + 9)*x)*v[i - 1] + (5 + 18*x + 9*x^2)*deriv(v[i - 1], x))/(6*n - 12)); 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-976cdb6e4a693ec8","steps":["basis A013610","D"],"edges":["e-1c0d9915253416a6","e-ac8bcc7f7f30472d"],"text":"A078812 \u2192 basis A013610 \u2192 D"},"headline":"A078812 \u2192 basis A013610 \u2192 D","chain":null}