{"id":"n-2abd14fc158c8140","revision":0,"first_index":0,"fingerprint":"f6-1920365dcfdf3eae","name":"theta_plus_one of basis_map_A013610 of A Jacobsthal triangle.","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"],["1","6"],["3","24","27"],["5","66","189","108"],["11","180","756","1080","405"],["21","438","2538","5832","5265","1458"],["43","1056","7641","24192","36045","23328","5103"],["85","2442","21411","86724","181035","193914","96957","17496"],["171","5580","57186","281880","759780","1151820","949158","384912","59049"],["341","12510","147096","855360","2829330","5578308","6531840","4339008","1476225","196830"],["683","27768","367983","2464992","9670590","23537952","36016974","34012224","18836631","5511240","649539"],["1365","60978","899721","6824412","30988170","89879868","170511642","210966768","165868641","78535170","20135709","2125764"],["2731","132900","2160000","18297144","94408335","318141432","721890792","1110576096","1145137257","768030660","316975032","72275976","6908733"],["5461","287622","5106294","47783736","276182055","1060885998","2805568164","5176786464","6649094547","5848016130","3410079750","1245697704","255623121","22320522"],["10923","618960","11916693","122083200","781416315","3371164272","10189440765","21967137792","33907412025","37181974320","28411485399","14625256320","4787751969","892820880","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","54","368","2432","15552","97408","600064","3649536","21966848","131096576","776798208","4574937088","26803437568","156322234368"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["6","23","15"],["0","9","21"],["0","0","6"]]},"terms":[{"coeff":{"coeffs":[["30","142","186","90"],["-24","-20","126","126"],["0","-54","-54","36"],["0","0","-24","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["9","72","194","192","45"],["0","27","153","237","63"],["0","0","18","60","18"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179046,"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-a49ddbc8e33374d1"]},{"id":179047,"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-a49ddbc8e33374d1"]},{"id":179048,"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-a49ddbc8e33374d1"]}],"region_exact":{},"incoming_edges":[{"id":"e-a49ddbc8e33374d1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2abd14fc158c8140","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"f619ca4fb2c8ebeabe185598b4d1ecd9e1ce460665fac609db82c94d8892bb88","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-ad5020a0c2e8a31d","role":"sequence","name":"basis_map_A013610 of A Jacobsthal triangle.","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:   1,     6;\n  2:   3,    24,     27;\n  3:   5,    66,    189,    108;\n  4:  11,   180,    756,   1080,     405;\n  5:  21,   438,   2538,   5832,    5265,    1458;\n  6:  43,  1056,   7641,  24192,   36045,   23328,    5103;\n  7:  85,  2442,  21411,  86724,  181035,  193914,   96957,   17496;\n  8: 171,  5580,  57186, 281880,  759780, 1151820,  949158,  384912,   59049;\n  9: 341, 12510, 147096, 855360, 2829330, 5578308, 6531840, 4339008, 1476225, 196830;\n","recurrence_sentence":"The row polynomials satisfy (6*n+(9*n^2+23*n)*x+(6*n^3+21*n^2+15*n)*x^2)*P(n,x) = -(24*n-30+(54*n^2+20*n-142)*x+(24*n^3+54*n^2-126*n-186)*x^2-(36*n^2+126*n+90)*x^3)*P(n-1,x) + (9+(27*n+72)*x+(18*n^2+153*n+194)*x^2+(60*n^2+237*n+192)*x^3+(18*n^2+63*n+45)*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[(-(24*n - 30 + (54*n^2 + 20*n - 142)*x + (24*n^3 + 54*n^2 - 126*n - 186)*x^2 - (36*n^2 + 126*n + 90)*x^3)*P[n - 1, x] + (9 + (27*n + 72)*x + (18*n^2 + 153*n + 194)*x^2 + (60*n^2 + 237*n + 192)*x^3 + (18*n^2 + 63*n + 45)*x^4)*D[P[n - 1, x], x])/(6*n + (9*n^2 + 23*n)*x + (6*n^3 + 21*n^2 + 15*n)*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] = (-(24*n - 30 + (54*n^2 + 20*n - 142)*x + (24*n^3 + 54*n^2 - 126*n - 186)*x^2 - (36*n^2 + 126*n + 90)*x^3)*v[i - 1] + (9 + (27*n + 72)*x + (18*n^2 + 153*n + 194)*x^2 + (60*n^2 + 237*n + 192)*x^3 + (18*n^2 + 63*n + 45)*x^4)*deriv(v[i - 1], x))/(6*n + (9*n^2 + 23*n)*x + (6*n^3 + 21*n^2 + 15*n)*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-ba196dcd01800741","steps":["basis A013610","\u03b8+1"],"edges":["e-900137e7780f9cba","e-a49ddbc8e33374d1"],"text":"A113953 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A113953 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}