{"id":"n-8be0b20fc2f56554","revision":1,"first_index":1,"fingerprint":"f6-90c2a235bbf6b62e","name":"one_plus_derivative of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","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"],["3","2"],["7","12","3"],["15","50","30","4"],["31","180","195","60","5"],["63","602","1050","560","105","6"],["127","1932","5103","4200","1330","168","7"],["255","6050","23310","27804","13230","2772","252","8"],["511","18660","102315","170100","114135","35280","5250","360","9"],["1023","57002","437250","986920","897435","383922","83160","9240","495","10"],["2047","173052","1834503","5517600","6618260","3764376","1113189","178200","15345","660","11"],["4095","523250","7597590","30034004","46606560","34292544","13297284","2876016","353925","24310","858","12"],["8191","1577940","31175235","160300140","317181865","295975680","146386240","41081040","6774768","660660","37037","1092","13"],["16383","4750202","127067850","843067680","2103466365","2452447998","1516394880","537027920","113963850","14794780","1171170","54600","1365","14"],["32767","14283372","515396703","4384762200","13674632790","19691295624","14992348371","6566274000","1743794910","289369080","30328298","1987440","78260","1680","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","22","99","471","2386","12867","73681","446620","2856457","19217243","135610448","1001159901","7714225057","61904585510"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","2"],["7","12","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","2","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","1"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2","-5","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","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":165305,"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-8b35b367ffdd04e7"]},{"id":165306,"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-8b35b367ffdd04e7"]},{"id":165307,"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-8b35b367ffdd04e7"]}],"region_exact":{},"incoming_edges":[{"id":"e-8b35b367ffdd04e7","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-8be0b20fc2f56554","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":"f61246f10a84c2364354e3a97286313912eb6fe2f1bdac9f21ff7848e805814c","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-f25b32370a8d627e","role":"sequence","name":"Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","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":[{"id":"l-37fed09c3d316258","kind":"fingerprint_related","node_a_id":"n-d99a5c662672737d","node_b_id":"n-8be0b20fc2f56554","fingerprint":"f6-90c2a235bbf6b62e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90c2a235bbf6b62e"},"other_id":"n-d99a5c662672737d","other":{"id":"n-d99a5c662672737d","name":"derivative of Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-90c2a235bbf6b62e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["3","2"]],"card_rows":[["1"],["3","2"],["7","12","3"],["15","50","30","4"]],"row_sums_preview":["0","1","5","22","99","471","2386","12867"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["D"],"edges":["e-d6bdcc4fc26e5291"],"text":"A008277 \u2192 D"},"headline":"A008277 \u2192 D","chain":null}},{"id":"l-f0ab320a48f1a8e1","kind":"oeis_match","node_a_id":"n-8be0b20fc2f56554","node_b_id":null,"fingerprint":null,"oeis_id":"A033452","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A033452","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A033452","relation":"match_row_sums"}],"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:    3,     2;\n   3:    7,    12,      3;\n   4:   15,    50,     30,      4;\n   5:   31,   180,    195,     60,      5;\n   6:   63,   602,   1050,    560,    105,      6;\n   7:  127,  1932,   5103,   4200,   1330,    168,     7;\n   8:  255,  6050,  23310,  27804,  13230,   2772,   252,    8;\n   9:  511, 18660, 102315, 170100, 114135,  35280,  5250,  360,   9;\n  10: 1023, 57002, 437250, 986920, 897435, 383922, 83160, 9240, 495, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*x+2*x^2+x^3)*P(n-3,x) + (x^2+x^3)*d/dx P(n-3,x) - (2+5*x+3*x^2)*P(n-2,x) - (x+2*x^2)*d/dx P(n-2,x) + (3+3*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 7 + 12*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[3, x_] := 7 + 12*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(2*x + 2*x^2 + x^3)*P[n - 3, x] + (x^2 + x^3)*D[P[n - 3, x], x] - (2 + 5*x + 3*x^2)*P[n - 2, x] - (x + 2*x^2)*D[P[n - 2, x], x] + (3 + 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] = 3 + 2*x; v[3] = 7 + 12*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (2*x + 2*x^2 + x^3)*v[i - 3] + (x^2 + x^3)*deriv(v[i - 3], x) - (2 + 5*x + 3*x^2)*v[i - 2] - (x + 2*x^2)*deriv(v[i - 2], x) + (3 + 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-3cb22164e2f83fe1","steps":["\u03b8+1","D+1"],"edges":["e-907e3c6c6275331d","e-8b35b367ffdd04e7"],"text":"A008277 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A008277 \u2192 \u03b8+1 \u2192 D+1","chain":null}
