{"id":"n-e066b43456aa872f","revision":0,"first_index":1,"fingerprint":"f6-53291945d5eef4b9","name":"derivative of veronese_section_2_0 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["5"],["35"],["126","18"],["330","330"],["715","2574","39"],["1365","12870","1365"],["2380","48620","18564","68"],["3876","151164","151164","3876"],["5985","406980","881790","81396","105"],["8855","980628","4056234","980628","8855"],["12650","2163150","15600900","8171900","265650","150"],["17550","4440150","52151580","52151580","4440150","17550"],["23751","8584290","155687805","271455660","50075025","712530","203"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","5","35","144","660","3328","15600","69632","310080","1376256","6035200","26214400","113218560","486539264"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["5"],["35"],["126","18"],["330","330"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["191","-381","189","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-64","192","-192","64"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["3342","-2252","-66"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["-128","1264","-1120","-16"]]},"deriv_order":1,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["9041","146","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1152","5184","64"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1860","52"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2048","416","32"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-111","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1472","128"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["14"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-384","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176513,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-93dc3f1973b8ef22"]},{"id":176514,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-93dc3f1973b8ef22"]},{"id":176515,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-93dc3f1973b8ef22"]}],"region_exact":{},"incoming_edges":[{"id":"e-93dc3f1973b8ef22","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-e066b43456aa872f","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":"284ad75b513809a865a62fb812902258971f82e97966783b82a92ab42133d353","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6a79351bf6797e1b","role":"sequence","name":"veronese_section_2_0 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-108bc0bb1037ff4a","kind":"fingerprint_related","node_a_id":"n-3c595677b0c45eaa","node_b_id":"n-e066b43456aa872f","fingerprint":"f6-53291945d5eef4b9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-53291945d5eef4b9"},"other_id":"n-3c595677b0c45eaa","other":{"id":"n-3c595677b0c45eaa","name":"veronese_section_2_1 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-53291945d5eef4b9","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["10"]],"card_rows":[["2"],["10"],["28","4"],["60","60"]],"row_sums_preview":["0","2","10","32","120","512","2080","8192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["Veronese 2 (j=1)"],"edges":["e-7007453d8749332a"],"text":"A091044 \u2192 Veronese 2 (j=1)"},"headline":"A091044 \u2192 Veronese 2 (j=1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    0;\n   2:    0;\n   3:    5;\n   4:   35;\n   5:  126,     18;\n   6:  330,    330;\n   7:  715,   2574,     39;\n   8: 1365,  12870,   1365;\n   9: 2380,  48620,  18564,   68;\n  10: 3876, 151164, 151164, 3876;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (191-381*x+189*x^2+x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3342-2252*x-66*x^2)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (9041+146*x-3*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) + (1860+52*x)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) - (111-3*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) + 14*P(n-1,x) - (384+16*x)*d/dx P(n-1,x) for n >= 7, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 5, P(4,x) = 35, P(5,x) = 126 + 18*x, P(6,x) = 330 + 330*x.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 5;\nP[4, x_] := 35;\nP[5, x_] := 126 + 18*x;\nP[6, x_] := 330 + 330*x;\nP[n_, x_] := P[n, x] = Expand[(191 - 381*x + 189*x^2 + x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3342 - 2252*x - 66*x^2)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (9041 + 146*x - 3*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] + (1860 + 52*x)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] - (111 - 3*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] + 14*P[n - 1, x] - (384 + 16*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 5; v[4] = 35; v[5] = 126 + 18*x; v[6] = 330 + 330*x; for(n = 7, N, my(i = n); v[i] = (191 - 381*x + 189*x^2 + x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3342 - 2252*x - 66*x^2)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (9041 + 146*x - 3*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) + (1860 + 52*x)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) - (111 - 3*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) + 14*v[i - 1] - (384 + 16*x)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["Veronese 2","D"],"edges":["e-d03002d6b7efed5a","e-93dc3f1973b8ef22"],"text":"A091042 \u2192 Veronese 2 \u2192 D"},"headline":"A091042 \u2192 Veronese 2 \u2192 D","chain":null}
