{"id":"n-28f29d8563043b26","revision":1,"first_index":0,"fingerprint":"f6-2c994ef0f89cedbf","name":"derivative of Stirling2 triangle with scaled diagonals (powers of 2).","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"],["6","2"],["28","24","3"],["120","200","60","4"],["496","1440","780","120","5"],["2016","9632","8400","2240","210","6"],["8128","61824","81648","33600","5320","336","7"],["32640","387200","745920","444864","105840","11088","504","8"],["130816","2388480","6548160","5443200","1826160","282240","21000","720","9"],["523776","14592512","55968000","63162880","28717920","6142752","665280","36960","990","10"],["2096128","88602624","469632768","706252800","423568640","120460032","17811024","1425600","61380","1320","11"],["8386560","535808000","3889966080","7688705024","5965639680","2194722816","425513088","46016256","2831400","97240","1716","12"],["33550336","3231621120","31923440640","82073671680","81198557440","37884887040","9368719360","1314593280","108396288","5285280","148148","2184","13"],["134209536","19456827392","260234956800","863301304320","1076974778880","627826687488","194098544640","34369786880","3646843200","236716480","9369360","218400","2730","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","8","55","384","2841","22504","190863","1728064","16640785","169811080","1829912327","20757687872","247142872809","3080290246120"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165866,"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-775d8a1aaaf30263"]},{"id":165867,"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-775d8a1aaaf30263"]},{"id":165868,"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-775d8a1aaaf30263"]},{"id":165869,"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-775d8a1aaaf30263"]}],"region_exact":{},"incoming_edges":[{"id":"e-775d8a1aaaf30263","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-28f29d8563043b26","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":"baf9c6e85d9eabc834b6a3eab813f25b52bc197e1bb14ffb7d0260b510f3cc49","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-a7e9ebb9860a9dbe","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 2).","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:      1;\n  2:      6,       2;\n  3:     28,      24,       3;\n  4:    120,     200,      60,       4;\n  5:    496,    1440,     780,     120,       5;\n  6:   2016,    9632,    8400,    2240,     210,      6;\n  7:   8128,   61824,   81648,   33600,    5320,    336,     7;\n  8:  32640,  387200,  745920,  444864,  105840,  11088,   504,   8;\n  9: 130816, 2388480, 6548160, 5443200, 1826160, 282240, 21000, 720, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8+4*x+x^2)*P(n-2,x) - (4*x+2*x^2)*d/dx P(n-2,x) + (6+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(8 + 4*x + x^2)*P[n - 2, x] - (4*x + 2*x^2)*D[P[n - 2, x], x] + (6 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 4*x + x^2)*v[i - 2] - (4*x + 2*x^2)*deriv(v[i - 2], x) + (6 + 2*x)*v[i - 1] + 2*x*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a7e9ebb9860a9dbe","steps":["D"],"edges":["e-775d8a1aaaf30263"],"text":"A075497 \u2192 D"},"headline":"A075497 \u2192 D","chain":null}