{"id":"n-50fa68ec9dbe3daf","revision":0,"first_index":0,"fingerprint":"f6-c1ad9de40a35a36b","name":"theta_plus_one of Binomial coefficients C(n,k) with n-k odd, 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"],["2"],["1","6"],["4","8"],["1","20","15"],["6","40","18"],["1","42","105","28"],["8","112","168","32"],["1","72","378","336","45"],["10","240","756","480","50"],["1","110","990","1848","825","66"],["12","440","2376","3168","1100","72"],["1","156","2145","6864","6435","1716","91"],["14","728","6006","13728","10010","2184","98"],["1","210","4095","20020","32175","18018","3185","120"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","7","12","36","64","176","320","832","1536","3840","7168","17408","32768","77824"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["1","6"],["4","8"],["1","20","15"],["6","40","18"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-13","27","-15","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["4","-12","12","-4"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-23","10","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-8","0","8"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["5","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","-4"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171949,"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-ba9cbe1cebceeb56"]},{"id":171950,"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-ba9cbe1cebceeb56"]},{"id":171951,"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-ba9cbe1cebceeb56"]}],"region_exact":{},"incoming_edges":[{"id":"e-ba9cbe1cebceeb56","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-50fa68ec9dbe3daf","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":"598cd24b715bc90bf48c6ecf3892b67384feb2d3b18e9c6ca1ae09054b73bd46","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-632227a1ddc9eaa6","role":"sequence","name":"Binomial coefficients C(n,k) with n-k odd, 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:  2;\n  2:  1,   6;\n  3:  4,   8;\n  4:  1,  20,  15;\n  5:  6,  40,  18;\n  6:  1,  42, 105,  28;\n  7:  8, 112, 168,  32;\n  8:  1,  72, 378, 336, 45;\n  9: 10, 240, 756, 480, 50;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(13-27*x+15*x^2-x^3)*P(n-6,x) + (4-12*x+12*x^2-4*x^3)*d/dx P(n-6,x) - (23-10*x+3*x^2)*P(n-4,x) - (8-8*x^2)*d/dx P(n-4,x) + (5+3*x)*P(n-2,x) + (4-4*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 1 + 6*x, P(3,x) = 4 + 8*x, P(4,x) = 1 + 20*x + 15*x^2, P(5,x) = 6 + 40*x + 18*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 1 + 6*x;\nP[3, x_] := 4 + 8*x;\nP[4, x_] := 1 + 20*x + 15*x^2;\nP[5, x_] := 6 + 40*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[-(13 - 27*x + 15*x^2 - x^3)*P[n - 6, x] + (4 - 12*x + 12*x^2 - 4*x^3)*D[P[n - 6, x], x] - (23 - 10*x + 3*x^2)*P[n - 4, x] - (8 - 8*x^2)*D[P[n - 4, x], x] + (5 + 3*x)*P[n - 2, x] + (4 - 4*x)*D[P[n - 2, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 1 + 6*x; v[4] = 4 + 8*x; v[5] = 1 + 20*x + 15*x^2; v[6] = 6 + 40*x + 18*x^2; for(n = 6, N, my(i = n + 1); v[i] = -(13 - 27*x + 15*x^2 - x^3)*v[i - 6] + (4 - 12*x + 12*x^2 - 4*x^3)*deriv(v[i - 6], x) - (23 - 10*x + 3*x^2)*v[i - 4] - (8 - 8*x^2)*deriv(v[i - 4], x) + (5 + 3*x)*v[i - 2] + (4 - 4*x)*deriv(v[i - 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-e655e253465c7aff","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-c2ab5705bc1e09a1","e-ba9cbe1cebceeb56"],"text":"Veronese parent of A109447 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"Veronese parent of A109447 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}