{"id":"n-306ee888f543aefe","revision":1,"first_index":1,"fingerprint":"f6-03d469782620294b","name":"one_plus_derivative of Triangle of B-analogs of Stirling numbers of the second kind.","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"],["2","1"],["5","6","1"],["14","31","12","1"],["41","156","106","20","1"],["122","781","840","270","30","1"],["365","3906","6331","3100","575","42","1"],["1094","19531","46452","32711","9030","1085","56","1"],["3281","97656","335476","328440","126966","22344","1876","72","1"],["9842","488281","2398800","3195340","1666980","403662","49056","3036","90","1"],["29525","2441406","17040661","30437000","20913250","6704964","1107162","98280","4665","110","1"],["88574","12207031","120520092","285695421","254193060","105153070","22682352","2709762","183150","6875","132","1"],["265721","61035156","849788446","2653625260","3019933631","1582721448","434273092","67095864","6057447","321860","9790","156","1"],["797162","305175781","5979169560","24459281210","35277772530","23110350263","7913857952","1539000892","178288110","12576707","538824","13546","182","1"],["2391485","1525878906","42007173391","224170373700","406885540425","329782619166","138954778963","33355384920","4824285895","433943510","24557533","865956","18291","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","58","324","2044","14320","109960","916112","8215088","78777024","803439520","8675127872","98776822720","1181967812352"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165569,"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-4980121166fd7e9f"]},{"id":165570,"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-4980121166fd7e9f"]},{"id":165571,"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-4980121166fd7e9f"]},{"id":165572,"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-4980121166fd7e9f"]}],"region_exact":{},"incoming_edges":[{"id":"e-4980121166fd7e9f","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-306ee888f543aefe","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":"537fb8020f27c9ff60e117b37618d680e951cffcc2a503cf1e6734c72adf4e2e","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-d55b2cfe05a04774","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of the second kind.","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:    1;\n   2:    2,      1;\n   3:    5,      6,       1;\n   4:   14,     31,      12,       1;\n   5:   41,    156,     106,      20,       1;\n   6:  122,    781,     840,     270,      30,      1;\n   7:  365,   3906,    6331,    3100,     575,     42,     1;\n   8: 1094,  19531,   46452,   32711,    9030,   1085,    56,    1;\n   9: 3281,  97656,  335476,  328440,  126966,  22344,  1876,   72,  1;\n  10: 9842, 488281, 2398800, 3195340, 1666980, 403662, 49056, 3036, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3-x^2)*P(n-2,x) - (2*x-2*x^2)*d/dx P(n-2,x) + 4*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(3 - x^2)*P[n - 2, x] - (2*x - 2*x^2)*D[P[n - 2, x], x] + 4*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(3 - x^2)*v[i - 2] - (2*x - 2*x^2)*deriv(v[i - 2], x) + 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d55b2cfe05a04774","steps":["D+1"],"edges":["e-4980121166fd7e9f"],"text":"A039755 \u2192 D+1"},"headline":"A039755 \u2192 D+1","chain":null}