{"id":"n-8bc7d0e7da25f478","revision":1,"first_index":1,"fingerprint":"f6-932956b939518c2d","name":"derivative of Triangle read by rows: 3-Stirling numbers of the second kind.","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":[[],["1"],["7","2"],["37","24","3"],["175","194","54","4"],["781","1320","615","100","5"],["3367","8162","5670","1520","165","6"],["14197","47544","46263","18200","3220","252","7"],["58975","266114","348894","189084","48510","6132","364","8"],["242461","1448520","2492535","1788780","624435","113400","10794","504","9"],["989527","7727522","17127990","15844840","7231455","1769922","240240","17880","675","10"],["4017157","40616664","114359223","133751200","77657690","24607044","4467309","471240","28215","880","11"],["16245775","211117634","747080334","1088737364","788450520","314652624","73381308","10289136","868725","42790","1122","12"],["65514541","1088079720","4799158455","8617268660","7668525865","3778014240","1100907808","197044848","21999978","1521520","62777","1404","13"],["263652487","5571427682","30427070310","66719758560","72119792745","43204359198","15416761360","3425673680","485675190","44224180","2552550","89544","1729","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","9","64","427","2821","18890","129683","918081","6721438","50950061","399976633","3250867344","27338099829","237681039229"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-12","-6","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165840,"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-6eaf06ae1521efeb"]},{"id":165841,"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-6eaf06ae1521efeb"]},{"id":165842,"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-6eaf06ae1521efeb"]},{"id":165843,"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-6eaf06ae1521efeb"]}],"region_exact":{},"incoming_edges":[{"id":"e-6eaf06ae1521efeb","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-8bc7d0e7da25f478","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":"b033e13628fb2452070aa091e2f713488ab1f9137339cbc252503cd914cbce17","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-a2b8ff03da0c14a3","role":"sequence","name":"Triangle read by rows: 3-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:      0;\n   2:      1;\n   3:      7,       2;\n   4:     37,      24,       3;\n   5:    175,     194,      54,       4;\n   6:    781,    1320,     615,     100,      5;\n   7:   3367,    8162,    5670,    1520,    165,      6;\n   8:  14197,   47544,   46263,   18200,   3220,    252,     7;\n   9:  58975,  266114,  348894,  189084,  48510,   6132,   364,   8;\n  10: 242461, 1448520, 2492535, 1788780, 624435, 113400, 10794, 504, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(12+6*x+x^2)*P(n-2,x) - (3*x+x^2)*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(12 + 6*x + x^2)*P[n - 2, x] - (3*x + x^2)*D[P[n - 2, x], x] + (7 + 2*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(12 + 6*x + x^2)*v[i - 2] - (3*x + x^2)*deriv(v[i - 2], x) + (7 + 2*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-a2b8ff03da0c14a3","steps":["D"],"edges":["e-6eaf06ae1521efeb"],"text":"A143495 \u2192 D"},"headline":"A143495 \u2192 D","chain":null}