{"id":"n-a34e1bb09a598707","revision":0,"first_index":0,"fingerprint":"f6-09e472dd4f3384ea","name":"one_plus_derivative of reversal of Number triangle T(n,k)=C(n,k)C(2,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","2"],["5","6","1"],["7","12","3"],["9","20","6"],["11","30","10"],["13","42","15"],["15","56","21"],["17","72","28"],["19","90","36"],["21","110","45"],["23","132","55"],["25","156","66"],["27","182","78"],["29","210","91"]],"degrees":[0,1,2,2,2,2,2,2,2,2,2,2,2,2,2],"row_sums":["1","5","12","22","35","51","70","92","117","145","176","210","247","287","330"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"],["5","6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171272,"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-d384ce60c8a3c0b1"]},{"id":171273,"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-d384ce60c8a3c0b1"]},{"id":171274,"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-d384ce60c8a3c0b1"]}],"region_exact":{},"incoming_edges":[{"id":"e-d384ce60c8a3c0b1","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-a34e1bb09a598707","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":"9253b4ba6085250ce23dbee7602e2cb6f39d19d46ef777851aff74f424e02b78","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-88b9d8361c4b11db","role":"sequence","name":"reversal of Number triangle T(n,k)=C(n,k)C(2,n-k).","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:  1;\n  1:  3,  2;\n  2:  5,  6,  1;\n  3:  7, 12,  3;\n  4:  9, 20,  6;\n  5: 11, 30, 10;\n  6: 13, 42, 15;\n  7: 15, 56, 21;\n  8: 17, 72, 28;\n  9: 19, 90, 36;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-3,x) - 3*P(n-2,x) + 3*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 5 + 6*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[2, x_] := 5 + 6*x + x^2;\nP[n_, x_] := P[n, x] = Expand[P[n - 3, x] - 3*P[n - 2, x] + 3*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 5 + 6*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = v[i - 3] - 3*v[i - 2] + 3*v[i - 1]); 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-1815d3e1c9c57d5b","steps":["reverse","D+1"],"edges":["e-29732eeeac05382b","e-d384ce60c8a3c0b1"],"text":"A106246 \u2192 reverse \u2192 D+1"},"headline":"A106246 \u2192 reverse \u2192 D+1","chain":null}