{"id":"n-d7e00a1c4968fb6d","revision":1,"first_index":0,"fingerprint":"f6-d79691bcc7e60ca0","name":"derivative of Triangular array of Motzkin polynomial coefficients.","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":[[],[],["1"],["3"],["6","4"],["10","20"],["15","60","15"],["21","140","105"],["28","280","420","56"],["36","504","1260","504"],["45","840","3150","2520","210"],["55","1320","6930","9240","2310"],["66","1980","13860","27720","13860","792"],["78","2860","25740","72072","60060","10296"],["91","4004","45045","168168","210210","72072","3003"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","3","10","30","90","266","784","2304","6765","19855","58278","171106","502593"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-4"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["2","-12"],["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","2","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161245,"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-07f016342004b27e"]},{"id":161246,"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-07f016342004b27e"]},{"id":161247,"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-07f016342004b27e"]}],"region_exact":{},"incoming_edges":[{"id":"e-07f016342004b27e","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-d7e00a1c4968fb6d","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":"b552788f07588987d474c1adbfcfc51a2c689b9100f7f973b366e63b34658aa4","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-169bd483754ab390","role":"sequence","name":"Triangular array of Motzkin polynomial coefficients.","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-b9a2152a13f0bfd0","kind":"oeis_match","node_a_id":"n-d7e00a1c4968fb6d","node_b_id":null,"fingerprint":null,"oeis_id":"A014531","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A014531","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A014531","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-d7e00a1c4968fb6d","oeis_id":"A014531","target":"row_sums","oeis_offset":1,"first_row":2,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":10421}],"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:  1;\n  3:  3;\n  4:  6,   4;\n  5: 10,  20;\n  6: 15,  60,   15;\n  7: 21, 140,  105;\n  8: 28, 280,  420,  56;\n  9: 36, 504, 1260, 504;\n","recurrence_sentence":"The row polynomials satisfy (n^2-4)*P(n,x) = (n^2+2*n+(4*n^2-12*n)*x)*P(n-1,x) + (2*n*x-8*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + (4*n^2 - 12*n)*x)*P[n - 1, x] + (2*n*x - 8*n*x^2)*D[P[n - 1, x], x])/(n^2 - 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] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 + 2*n + (4*n^2 - 12*n)*x)*v[i - 1] + (2*n*x - 8*n*x^2)*deriv(v[i - 1], x))/(n^2 - 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-2a05d8340c329b21","steps":["\u03b3 (shift +1)","D"],"edges":["e-3261655b8c5a1522","e-07f016342004b27e"],"text":"narayana \u2192 \u03b3 (shift +1) \u2192 D"},"headline":"narayana \u2192 \u03b3 (shift +1) \u2192 D","chain":null}