{"id":"n-71837b9dafc9839a","revision":1,"first_index":0,"fingerprint":"f6-03590a173d64a7af","name":"one_plus_derivative of A133080 * A007318.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["3","4","1"],["7","13","7","1"],["5","16","18","8","1"],["11","41","58","38","11","1"],["7","36","75","80","45","12","1"],["15","85","201","255","185","75","15","1"],["9","64","196","336","350","224","84","16","1"],["19","145","484","924","1106","854","420","124","19","1"],["11","100","405","960","1470","1512","1050","480","135","20","1"],["23","221","955","2445","4110","4746","3822","2130","795","185","23","1"],["13","144","726","2200","4455","6336","6468","4752","2475","880","198","24","1"],["27","313","1662","5346","11605","17919","20196","16764","10197","4455","1342","258","27","1"],["15","196","1183","4368","11011","20020","27027","27456","21021","12012","5005","1456","273","28","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","8","28","48","160","256","832","1280","4096","6144","19456","28672","90112","131072"],"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":[["-1","-2","1"],["-1","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["-4","-11","2","3"],["-5","13","5","-2"],["9","-1","-4","0"],["-3","-2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-3","1","5","2"],["-1","5","10","4","0"],["3","5","2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161148,"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-d70a0ba3fa52a1a1"]},{"id":161149,"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-d70a0ba3fa52a1a1"]},{"id":161150,"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-d70a0ba3fa52a1a1"]}],"region_exact":{},"incoming_edges":[{"id":"e-d70a0ba3fa52a1a1","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-71837b9dafc9839a","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":"28daa168c3114c7d8869362a7e974943473af06bb376bebc25d95a6ec90eaf35","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-d75669a8cd7f9393","role":"sequence","name":"A133080 * A007318.","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":[],"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:  3,   1;\n  2:  3,   4,   1;\n  3:  7,  13,   7,   1;\n  4:  5,  16,  18,   8,    1;\n  5: 11,  41,  58,  38,   11,   1;\n  6:  7,  36,  75,  80,   45,  12,   1;\n  7: 15,  85, 201, 255,  185,  75,  15,   1;\n  8:  9,  64, 196, 336,  350, 224,  84,  16,  1;\n  9: 19, 145, 484, 924, 1106, 854, 420, 124, 19, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-1+(2*n-2)*x+x^2)*P(n,x) = -(3*n^3-9*n^2+5*n+4+(2*n^3+n^2-13*n+11)*x+(4*n^2-5*n-2)*x^2+(2*n-3)*x^3)*P(n-1,x) + (3*n^2-n-1+(5*n^2+5*n-3)*x+(2*n^2+10*n+1)*x^2+(4*n+5)*x^3+2*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[(-(3*n^3 - 9*n^2 + 5*n + 4 + (2*n^3 + n^2 - 13*n + 11)*x + (4*n^2 - 5*n - 2)*x^2 + (2*n - 3)*x^3)*P[n - 1, x] + (3*n^2 - n - 1 + (5*n^2 + 5*n - 3)*x + (2*n^2 + 10*n + 1)*x^2 + (4*n + 5)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^2 - n - 1 + (2*n - 2)*x + 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] = (-(3*n^3 - 9*n^2 + 5*n + 4 + (2*n^3 + n^2 - 13*n + 11)*x + (4*n^2 - 5*n - 2)*x^2 + (2*n - 3)*x^3)*v[i - 1] + (3*n^2 - n - 1 + (5*n^2 + 5*n - 3)*x + (2*n^2 + 10*n + 1)*x^2 + (4*n + 5)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^2 - n - 1 + (2*n - 2)*x + 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-d75669a8cd7f9393","steps":["D+1"],"edges":["e-d70a0ba3fa52a1a1"],"text":"A133087 \u2192 D+1"},"headline":"A133087 \u2192 D+1","chain":null}