{"id":"n-734c2ac4a375b77a","revision":0,"first_index":0,"fingerprint":"f6-695778c525463e73","name":"one_plus_derivative of Stirling2 triangle with scaled diagonals (powers of 4).","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":[["28","14","1"],["176","160","27","1"],["1216","1760","520","44","1"],["8960","19456","8880","1280","65","1"],["68608","218624","144256","31360","2660","90","1"],["536576","2498560","2295552","704256","88480","4928","119","1"],["4243456","28958720","36259840","15074304","2626176","213696","8400","152","1"],["33751040","339214336","571944960","313876480","72764160","8101632","460320","13440","189","1"],["269221888","4003856384","9031745536","6430392320","1929579520","282078720","21702912","908160","20460","230","1"],["2150629376","47510978560","142908260352","130475687936","49708482560","9276399616","927429888","52115712","1671120","29920","275","1"],["17192452096","565843066880","2265996328960","2632196030464","1255679000576","293182636032","37026795520","2681462784","114683712","2905760","42328","324","1"],["137489285120","6755782230016","35998783242240","52918254305280","31292260679680","9006631682048","1407808962560","127723847680","6992713728","234986752","4820816","58240","377","1"],["1099712954368","80794609516544","572808774025216","1061674266132480","772416220692480","271005849354240","51631885713408","5748896890880","391657277440","16754017280","453701248","7687680","78260","434","1"],["8796898328576","967336693596160","9126192246423552","21273332905148416","18936648161034240","8029733350539264","1842864376184832","247944482193408","20590313103360","1089934376960","37395438080","833018368","11851840","103040","495","1"],["70371965403136","11590449747722240","145548601332858880","425945928537473024","461959619544088576","235165846924689408","64420415888424960","10347777788215296","1031213818871808","66166564372480","2796021223424","78594926592","1464922368","17745280","133280","560","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["43","364","3541","38642","465599","6128472","87384745","1340126558","21969506131","383011685316","7069915405437","137651966814538","2817589088041959","60454567601340592","1356149268211071313"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["28","14","1"],["176","160","27","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-32","8","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-16","12"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189778,"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-74745f25c3d77bb6"]},{"id":189779,"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-74745f25c3d77bb6"]},{"id":189780,"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-74745f25c3d77bb6"]},{"id":189781,"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-74745f25c3d77bb6"]}],"region_exact":{},"incoming_edges":[{"id":"e-74745f25c3d77bb6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-734c2ac4a375b77a","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":"51eea349a1f0ae122dd02a3d6219d432f130acb008dc2a14751d849b613fced3","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-b17a463cc18bc314","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 4).","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:         28,          14,            1;\n  1:        176,         160,           27,            1;\n  2:       1216,        1760,          520,           44,           1;\n  3:       8960,       19456,         8880,         1280,          65,          1;\n  4:      68608,      218624,       144256,        31360,        2660,         90,         1;\n  5:     536576,     2498560,      2295552,       704256,       88480,       4928,       119,        1;\n  6:    4243456,    28958720,     36259840,     15074304,     2626176,     213696,      8400,      152,       1;\n  7:   33751040,   339214336,    571944960,    313876480,    72764160,    8101632,    460320,    13440,     189,     1;\n  8:  269221888,  4003856384,   9031745536,   6430392320,  1929579520,  282078720,  21702912,   908160,   20460,   230,   1;\n  9: 2150629376, 47510978560, 142908260352, 130475687936, 49708482560, 9276399616, 927429888, 52115712, 1671120, 29920, 275, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(32-8*x-3*x^2)*P(n-2,x) - (16*x-12*x^2)*d/dx P(n-2,x) + (12-2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 28 + 14*x + x^2, P(1,x) = 176 + 160*x + 27*x^2 + x^3.","mathematica":"P[0, x_] := 28 + 14*x + x^2;\nP[1, x_] := 176 + 160*x + 27*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(32 - 8*x - 3*x^2)*P[n - 2, x] - (16*x - 12*x^2)*D[P[n - 2, x], x] + (12 - 2*x)*P[n - 1, x] + 4*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] = 28 + 14*x + x^2; v[2] = 176 + 160*x + 27*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(32 - 8*x - 3*x^2)*v[i - 2] - (16*x - 12*x^2)*deriv(v[i - 2], x) + (12 - 2*x)*v[i - 1] + 4*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-b17a463cc18bc314","steps":["D+1"],"edges":["e-74745f25c3d77bb6"],"text":"A075499 \u2192 D+1"},"headline":"A075499 \u2192 D+1","chain":null}