{"id":"n-4523d7f1e2dcc56e","revision":0,"first_index":0,"fingerprint":"f6-2e1dc9a0697574fa","name":"derivative of veronese_section_2_0 of Triangle of coefficients in expansion of (2+3x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["9"],["54"],["216","162"],["720","1620"],["2160","9720","2187"],["6048","45360","30618"],["16128","181440","244944","26244"],["41472","653184","1469664","472392"],["103680","2177280","7348320","4723920","295245"],["253440","6842880","32332608","34642080","6495390"],["608256","20528640","129330432","207852480","77944680","3188646"],["1437696","59304960","480370176","1080832896","675520560","82904796"],["3354624","166053888","1681295616","5043886848","4728643920","1160667144","33480783"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","9","54","378","2340","14067","82026","468756","2636712","14648445","80566398","439453134","2380371084","12817382823"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["9"]],"recurrence":{"denominator":{"coeffs":[["4"],["-6"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-4","-27"],["4","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","8","-18"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178292,"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-eec8d935f90f7b86"]},{"id":178293,"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-eec8d935f90f7b86"]},{"id":178294,"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-eec8d935f90f7b86"]}],"region_exact":{},"incoming_edges":[{"id":"e-eec8d935f90f7b86","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-4523d7f1e2dcc56e","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":"22499387ae0ab5265ce4c7fcbb6aad002746921083d0400311475dcf7ef47be8","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-f2968a7a3d3a43cd","role":"sequence","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (2+3x)^n.","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-c159461abb7214be","kind":"fingerprint_related","node_a_id":"n-ec6cfd69d11c8345","node_b_id":"n-4523d7f1e2dcc56e","fingerprint":"f6-2e1dc9a0697574fa","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2e1dc9a0697574fa"},"other_id":"n-ec6cfd69d11c8345","other":{"id":"n-ec6cfd69d11c8345","name":"veronese_section_2_1 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-2e1dc9a0697574fa","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["12"]],"card_rows":[["3"],["12"],["36","27"],["96","216"]],"row_sums_preview":["0","3","12","63","312","1563","7812","39063"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["Veronese 2 (j=1)"],"edges":["e-396a3b284da38893"],"text":"A013620 \u2192 Veronese 2 (j=1)"},"headline":"A013620 \u2192 Veronese 2 (j=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:     0;\n  1:     0;\n  2:     9;\n  3:    54;\n  4:   216,    162;\n  5:   720,   1620;\n  6:  2160,   9720,    2187;\n  7:  6048,  45360,   30618;\n  8: 16128, 181440,  244944,  26244;\n  9: 41472, 653184, 1469664, 472392;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-6*n+4)*P(n,x) = (4*n^2-4*n+(9*n^2-27*n)*x)*P(n-1,x) + (8*n*x-18*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 9.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 9;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 4*n + (9*n^2 - 27*n)*x)*P[n - 1, x] + (8*n*x - 18*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 9; for(n = 3, N, my(i = n + 1); v[i] = ((4*n^2 - 4*n + (9*n^2 - 27*n)*x)*v[i - 1] + (8*n*x - 18*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4)); 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-5733d79f45a14084","steps":["Veronese 2","D"],"edges":["e-bdf59334244a08a5","e-eec8d935f90f7b86"],"text":"A013620 \u2192 Veronese 2 \u2192 D"},"headline":"A013620 \u2192 Veronese 2 \u2192 D","chain":null}