{"id":"n-2acd092e2d70afe2","revision":1,"first_index":0,"fingerprint":"f6-8ddb3dd77c8bf89a","name":"theta_plus_one of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","6"],["1","12"],["1","18","27"],["1","24","81"],["1","30","162","108"],["1","36","270","432"],["1","42","405","1080","405"],["1","48","567","2160","2025"],["1","54","756","3780","6075","1458"],["1","60","972","6048","14175","8748"],["1","66","1215","9072","28350","30618","5103"],["1","72","1485","12960","51030","81648","35721"],["1","78","1782","17820","85050","183708","142884","17496"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","13","46","106","301","739","1933","4801","12124","30004","74425","182917","448819"],"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"],["1","0"],["0","3"]]},"terms":[{"coeff":{"coeffs":[["0","-12","-36"],["1","12","18"],["0","3","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-18","-72"],["0","0","-3","-36"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163355,"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-825bd9c47e6b8dee"]},{"id":163356,"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-825bd9c47e6b8dee"]},{"id":163357,"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-825bd9c47e6b8dee"]}],"region_exact":{},"incoming_edges":[{"id":"e-825bd9c47e6b8dee","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-2acd092e2d70afe2","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":"a8054cbcd5e0a7dac65e0fa7386063463dce09360b934679a8ae29ba6dbfa4aa","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-e189038e95d51606","role":"sequence","name":"Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.","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;\n  2: 1,  6;\n  3: 1, 12;\n  4: 1, 18,  27;\n  5: 1, 24,  81;\n  6: 1, 30, 162,  108;\n  7: 1, 36, 270,  432;\n  8: 1, 42, 405, 1080,  405;\n  9: 1, 48, 567, 2160, 2025;\n","recurrence_sentence":"The row polynomials satisfy (n+3*n^2*x)*P(n,x) = (n+(3*n^2+12*n-12)*x+(18*n^2+18*n-36)*x^2)*P(n-1,x) - (x+(3*n+18)*x^2+(36*n+72)*x^3)*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[((n + (3*n^2 + 12*n - 12)*x + (18*n^2 + 18*n - 36)*x^2)*P[n - 1, x] - (x + (3*n + 18)*x^2 + (36*n + 72)*x^3)*D[P[n - 1, x], x])/(n + 3*n^2*x)]];\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] = ((n + (3*n^2 + 12*n - 12)*x + (18*n^2 + 18*n - 36)*x^2)*v[i - 1] - (x + (3*n + 18)*x^2 + (36*n + 72)*x^3)*deriv(v[i - 1], x))/(n + 3*n^2*x)); 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-e189038e95d51606","steps":["\u03b8+1"],"edges":["e-825bd9c47e6b8dee"],"text":"A304236 \u2192 \u03b8+1"},"headline":"A304236 \u2192 \u03b8+1","chain":null}