{"id":"n-8719fd1736db79b1","revision":0,"first_index":1,"fingerprint":"f6-aa827404435f1c6b","name":"one_plus_derivative of theta_plus_one of Triangle read by rows: 2-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"],["4","2"],["14","16","3"],["46","92","39","4"],["146","460","333","76","5"],["454","2132","2355","900","130","6"],["1394","9436","14973","8540","2035","204","7"],["4246","40532","89019","71204","24920","4074","301","8"],["12866","170620","505893","545916","263445","62496","7462","424","9"],["38854","708212","2785395","3950020","2526510","816102","140070","12768","576","10"],["117074","2910556","14989293","27413980","22623095","9560628","2212749","287760","20700","760","11"],["352246","11876372","79312299","184469604","192629140","103632738","31068807","5406456","551430","32120","979","12"],["1058786","48209980","414376053","1212567356","1578990985","1060216872","400524124","89571768","12147993","998140","48059","1236","13"],["3180454","194957492","2144195235","7827905540","12568646090","10376524158","4841636800","1353459536","234396162","25475450","1722721","69732","1534","14"],["9549554","786190876","11013412413","49826360220","97765404555","98094568572","55680676051","19067894560","4100999760","566145580","50418368","2855580","98553","1876","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","33","181","1020","5977","36589","234304","1569131","10978517","80136606","609332203","4818711365","39572170918","336964576533"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"],["14","16","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","10","5","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","2","3","1"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-11","-11","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178943,"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-bff2bd5be11643e0"]},{"id":178944,"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-bff2bd5be11643e0"]},{"id":178945,"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-bff2bd5be11643e0"]}],"region_exact":{},"incoming_edges":[{"id":"e-bff2bd5be11643e0","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-8719fd1736db79b1","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":"ec22db728a1ca6cce4bd685b83ab80050a4d36181d2d7c55ddc9cc6733df5114","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-62c0157c67ca5559","role":"sequence","name":"theta_plus_one of Triangle read by rows: 2-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:     4,      2;\n   3:    14,     16,       3;\n   4:    46,     92,      39,       4;\n   5:   146,    460,     333,      76,       5;\n   6:   454,   2132,    2355,     900,     130,      6;\n   7:  1394,   9436,   14973,    8540,    2035,    204,      7;\n   8:  4246,  40532,   89019,   71204,   24920,   4074,    301,     8;\n   9: 12866, 170620,  505893,  545916,  263445,  62496,   7462,   424,   9;\n  10: 38854, 708212, 2785395, 3950020, 2526510, 816102, 140070, 12768, 576, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+10*x+5*x^2+x^3)*P(n-3,x) + (2*x+3*x^2+x^3)*d/dx P(n-3,x) - (11+11*x+3*x^2)*P(n-2,x) - (3*x+2*x^2)*d/dx P(n-2,x) + (6+3*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 4 + 2*x, P(3,x) = 14 + 16*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[3, x_] := 14 + 16*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(6 + 10*x + 5*x^2 + x^3)*P[n - 3, x] + (2*x + 3*x^2 + x^3)*D[P[n - 3, x], x] - (11 + 11*x + 3*x^2)*P[n - 2, x] - (3*x + 2*x^2)*D[P[n - 2, x], x] + (6 + 3*x)*P[n - 1, x] + 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] = 4 + 2*x; v[3] = 14 + 16*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (6 + 10*x + 5*x^2 + x^3)*v[i - 3] + (2*x + 3*x^2 + x^3)*deriv(v[i - 3], x) - (11 + 11*x + 3*x^2)*v[i - 2] - (3*x + 2*x^2)*deriv(v[i - 2], x) + (6 + 3*x)*v[i - 1] + 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-6aaec68d653cbdb0","steps":["\u03b8+1","D+1"],"edges":["e-8098d1403de01053","e-bff2bd5be11643e0"],"text":"A143494 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A143494 \u2192 \u03b8+1 \u2192 D+1","chain":null}