{"id":"n-d2647fb576728480","revision":0,"first_index":0,"fingerprint":"f6-8f443bd722130c9e","name":"derivative of basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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":[[],[],["6"],["18"],["108","216"],["420","1080"],["2250","9720","9720"],["10206","52920","68040"],["55608","378000","816480","544320"],["281448","2204496","5715360","4898880"],["1602990","15014160","51030000","73483200","36741600"],["8839050","92877840","363741840","628689600","404157600"],["52925796","634784040","2972803680","6735960000","7274836800","2909934720"],["313256268","4136675400","21733414560","56743727040","73556683200","37829151360"],["1970881458","28897484616","173296042920","541050269760","919458540000","794412178560","264804059520"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","6","18","324","1500","21690","131166","1794408","13100184","177871950","1498305930","20581245036","194312907828","2723889456834"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-6"],["3"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-15","-54"],["6","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["-4","-24","-36"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180247,"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-262878b8d842f2bd"]},{"id":180248,"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-262878b8d842f2bd"]},{"id":180249,"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-262878b8d842f2bd"]}],"region_exact":{},"incoming_edges":[{"id":"e-262878b8d842f2bd","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-d2647fb576728480","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":"fc0cc91f986ceac17e931a38bae690a9315b58385124ae119eba55f3e1d1f6aa","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-b2d4a92113cb2b7b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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-316252da90518daa","kind":"fingerprint_related","node_a_id":"n-b2d4a92113cb2b7b","node_b_id":"n-d2647fb576728480","fingerprint":"f6-8f443bd722130c9e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8f443bd722130c9e"},"other_id":"n-b2d4a92113cb2b7b","other":{"id":"n-b2d4a92113cb2b7b","name":"basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","first_index":0,"fingerprint":"f6-8f443bd722130c9e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","6"]],"card_rows":[["1"],["1"],["3","6"],["7","18"]],"row_sums_preview":["1","1","9","25","241","1041","10681","60649"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c914b69838ac70ec","steps":["basis A013610"],"edges":["e-3c0217934a464502"],"text":"A059344 \u2192 basis A013610"},"headline":"A059344 \u2192 basis A013610","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:      6;\n  3:     18;\n  4:    108,     216;\n  5:    420,    1080;\n  6:   2250,    9720,    9720;\n  7:  10206,   52920,   68040;\n  8:  55608,  378000,  816480,  544320;\n  9: 281448, 2204496, 5715360, 4898880;\n","recurrence_sentence":"The row polynomials satisfy (3*n-6)*P(n,x) = (6*n^2-15*n+(18*n^2-54*n)*x)*P(n-1,x) - (4*n+24*n*x+36*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 15*n + (18*n^2 - 54*n)*x)*P[n - 1, x] - (4*n + 24*n*x + 36*n*x^2)*D[P[n - 1, x], x])/(3*n - 6)]];\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] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((6*n^2 - 15*n + (18*n^2 - 54*n)*x)*v[i - 1] - (4*n + 24*n*x + 36*n*x^2)*deriv(v[i - 1], x))/(3*n - 6)); 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-c914b69838ac70ec","steps":["basis A013610","D"],"edges":["e-3c0217934a464502","e-262878b8d842f2bd"],"text":"A059344 \u2192 basis A013610 \u2192 D"},"headline":"A059344 \u2192 basis A013610 \u2192 D","chain":null}