{"id":"n-a589983e2eca5f67","revision":0,"first_index":0,"fingerprint":"f6-7ac04ff230617c4a","name":"one_plus_derivative of basis_map_A007318 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["7","3"],["13","6"],["46","45","9"],["106","120","27"],["301","474","216","27"],["739","1350","738","108"],["1933","4287","3213","918","81"],["4801","12120","10719","3780","405"],["12124","35055","37512","17955","3645","243"],["30004","96744","118746","67932","17415","1458"],["74425","266604","374868","259038","89505","13851","729"],["182917","718380","1130688","901368","375921","74844","5103"],["448819","1922115","3364686","3074760","1547910","413343","51030","2187"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","10","19","100","253","1018","2935","10432","31825","106534","332299","1079020","3389221","10824850"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-12","-5","12"],["4","12","0"],["3","0","0"]]},"terms":[{"coeff":{"coeffs":[["30","-12","-114","-72"],["-69","-89","54","72"],["13","90","72","0"],["21","18","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-91","-422","-715","-528","-144"],["-130","-406","-420","-144","0"],["-39","-75","-36","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173658,"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-a3b1a946fd335630"]},{"id":173659,"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-a3b1a946fd335630"]},{"id":173660,"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-a3b1a946fd335630"]}],"region_exact":{},"incoming_edges":[{"id":"e-a3b1a946fd335630","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-a589983e2eca5f67","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":"dee6983cdfed0f533c1605537508f48a006fdb60f9993a8040c74eb8a5ed15f2","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-4d868f67dfcfb6b7","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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":[{"id":"l-b4e9ada126340d85","kind":"fingerprint_related","node_a_id":"n-a589983e2eca5f67","node_b_id":"n-b9aa6a5f16c078ec","fingerprint":"f6-7ac04ff230617c4a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7ac04ff230617c4a"},"other_id":"n-b9aa6a5f16c078ec","other":{"id":"n-b9aa6a5f16c078ec","name":"one_plus_derivative of basis_map_A013610 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","first_index":0,"fingerprint":"f6-7ac04ff230617c4a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["7","3"],["13","6"]],"card_rows":[["1"],["7","3"],["13","6"],["46","45","9"]],"row_sums_preview":["1","10","19","100","253","1018","2935","10432"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ac04e52510a48bb5","steps":["basis A013610","D+1"],"edges":["e-6895a0e432d3b559","e-404a39d061203adb"],"text":"A249139 \u2192 basis A013610 \u2192 D+1"},"headline":"A249139 \u2192 basis A013610 \u2192 D+1","chain":null}}],"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:    1;\n  2:    7,     3;\n  3:   13,     6;\n  4:   46,    45,     9;\n  5:  106,   120,    27;\n  6:  301,   474,   216,   27;\n  7:  739,  1350,   738,  108;\n  8: 1933,  4287,  3213,  918,  81;\n  9: 4801, 12120, 10719, 3780, 405;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3+4*n^2-12*n+(12*n^2-5*n)*x+12*n*x^2)*P(n,x) = (21*n^3+13*n^2-69*n+30+(18*n^3+90*n^2-89*n-12)*x+(72*n^2+54*n-114)*x^2+(72*n-72)*x^3)*P(n-1,x) - (39*n^2+130*n+91+(75*n^2+406*n+422)*x+(36*n^2+420*n+715)*x^2+(144*n+528)*x^3+144*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((21*n^3 + 13*n^2 - 69*n + 30 + (18*n^3 + 90*n^2 - 89*n - 12)*x + (72*n^2 + 54*n - 114)*x^2 + (72*n - 72)*x^3)*P[n - 1, x] - (39*n^2 + 130*n + 91 + (75*n^2 + 406*n + 422)*x + (36*n^2 + 420*n + 715)*x^2 + (144*n + 528)*x^3 + 144*x^4)*D[P[n - 1, x], x])/(3*n^3 + 4*n^2 - 12*n + (12*n^2 - 5*n)*x + 12*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((21*n^3 + 13*n^2 - 69*n + 30 + (18*n^3 + 90*n^2 - 89*n - 12)*x + (72*n^2 + 54*n - 114)*x^2 + (72*n - 72)*x^3)*v[i - 1] - (39*n^2 + 130*n + 91 + (75*n^2 + 406*n + 422)*x + (36*n^2 + 420*n + 715)*x^2 + (144*n + 528)*x^3 + 144*x^4)*deriv(v[i - 1], x))/(3*n^3 + 4*n^2 - 12*n + (12*n^2 - 5*n)*x + 12*n*x^2)); 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-e189038e95d51606","steps":["basis A007318","D+1"],"edges":["e-3091186ded9433db","e-a3b1a946fd335630"],"text":"A304236 \u2192 basis A007318 \u2192 D+1"},"headline":"A304236 \u2192 basis A007318 \u2192 D+1","chain":null}
