{"id":"n-286b3724a3c0878e","revision":0,"first_index":0,"fingerprint":"f6-ab01c2fc4dacda32","name":"theta_plus_one of veronese_section_2_0 of Even row Pascal-square read by antidiagonals.","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","20"],["1","42","15"],["1","72","105"],["1","110","378","28"],["1","156","990","336"],["1","210","2145","1848","45"],["1","272","4095","6864","825"],["1","342","7140","20020","6435","66"],["1","420","11628","49504","32175","1716"],["1","506","17955","108528","121550","18018","91"],["1","600","26565","217056","377910","116688","3185"],["1","702","37950","403788","1017450","554268","43316","120"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","21","58","178","517","1483","4249","12057","34004","95444","266649","742005","2057595"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1","6"],["1","20"],["1","42","15"],["1","72","105"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","12","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","4"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","-24","-15"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["0","0","-12","-4"]]},"deriv_order":1,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["12","3","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","12"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["27","10"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-4","-4","8"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-11","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","12"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173295,"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-4108b47711d467ef"]},{"id":173296,"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-4108b47711d467ef"]},{"id":173297,"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-4108b47711d467ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-4108b47711d467ef","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-286b3724a3c0878e","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":"63943e9be5c68f63d8eb9d87df1860e2f745d3a6b1c84efc5f8bd04a0abadcce","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-a11930116989dfbb","role":"sequence","name":"veronese_section_2_0 of Even row Pascal-square read by antidiagonals.","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,  20;\n  4: 1,  42,   15;\n  5: 1,  72,  105;\n  6: 1, 110,  378,   28;\n  7: 1, 156,  990,  336;\n  8: 1, 210, 2145, 1848,  45;\n  9: 1, 272, 4095, 6864, 825;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12*x^2+x^3)*P(n-6,x) + 4*x^3*d/dx P(n-6,x) - (24*x+15*x^2)*P(n-5,x) - (12*x^2+4*x^3)*d/dx P(n-5,x) + (12+3*x-3*x^2)*P(n-4,x) + 12*x*d/dx P(n-4,x) + (27+10*x)*P(n-3,x) - (4+4*x-8*x^2)*d/dx P(n-3,x) - (11-3*x)*P(n-2,x) + (8+12*x)*d/dx P(n-2,x) + 5*P(n-1,x) - (4+4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 6*x, P(3,x) = 1 + 20*x, P(4,x) = 1 + 42*x + 15*x^2, P(5,x) = 1 + 72*x + 105*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 6*x;\nP[3, x_] := 1 + 20*x;\nP[4, x_] := 1 + 42*x + 15*x^2;\nP[5, x_] := 1 + 72*x + 105*x^2;\nP[n_, x_] := P[n, x] = Expand[(12*x^2 + x^3)*P[n - 6, x] + 4*x^3*D[P[n - 6, x], x] - (24*x + 15*x^2)*P[n - 5, x] - (12*x^2 + 4*x^3)*D[P[n - 5, x], x] + (12 + 3*x - 3*x^2)*P[n - 4, x] + 12*x*D[P[n - 4, x], x] + (27 + 10*x)*P[n - 3, x] - (4 + 4*x - 8*x^2)*D[P[n - 3, x], x] - (11 - 3*x)*P[n - 2, x] + (8 + 12*x)*D[P[n - 2, x], x] + 5*P[n - 1, x] - (4 + 4*x)*D[P[n - 1, 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] = 1; v[3] = 1 + 6*x; v[4] = 1 + 20*x; v[5] = 1 + 42*x + 15*x^2; v[6] = 1 + 72*x + 105*x^2; for(n = 6, N, my(i = n + 1); v[i] = (12*x^2 + x^3)*v[i - 6] + 4*x^3*deriv(v[i - 6], x) - (24*x + 15*x^2)*v[i - 5] - (12*x^2 + 4*x^3)*deriv(v[i - 5], x) + (12 + 3*x - 3*x^2)*v[i - 4] + 12*x*deriv(v[i - 4], x) + (27 + 10*x)*v[i - 3] - (4 + 4*x - 8*x^2)*deriv(v[i - 3], x) - (11 - 3*x)*v[i - 2] + (8 + 12*x)*deriv(v[i - 2], x) + 5*v[i - 1] - (4 + 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-79ebafb266da3b41","steps":["Veronese 2","\u03b8+1"],"edges":["e-4815b3f40f31f82f","e-4108b47711d467ef"],"text":"A172431 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A172431 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}