{"id":"n-b2a2ce748aa32373","revision":0,"first_index":0,"fingerprint":"f6-d51a869d8526077f","name":"theta_plus_one of veronese_section_2_1 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["3"],["5","8"],["6","24"],["8","76","24"],["9","126","108"],["11","258","504","64"],["12","360","1080","384"],["14","608","3006","2368","160"],["15","780","5049","6240","1200"],["17","1180","10959","22000","9200","384"],["18","1440","16254","43344","28800","3456"],["20","2028","30384","115256","123000","31680","896"],["21","2394","41769","194556","278460","114912","9408"],["23","3206","70623","434180","877520","576576","100352","2048"],["24","3696","92232","665376","1663440","1475712","413952","24576"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["3","13","30","108","243","837","1836","6156","13284","43740","93312","303264","641520","2064528","4339008"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"],["5","8"],["6","24"],["8","76","24"],["9","126","108"],["11","258","504","64"]],"recurrence":{"denominator":{"coeffs":[["32"]]},"terms":[{"coeff":{"coeffs":[["-706","2280","-2208","256"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["81","-549","1044","-576"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-888","768","-384"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-162","-36","576"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["168","192"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["81","-144"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174059,"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-cca56a90043fa45b"]},{"id":174060,"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-cca56a90043fa45b"]},{"id":174061,"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-cca56a90043fa45b"]}],"region_exact":{},"incoming_edges":[{"id":"e-cca56a90043fa45b","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-b2a2ce748aa32373","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":"c989dfa31b21d9a9a1a1204153be23ce7f4d0664327125483802ce23ba8d3735","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-cfed7b5cee98d33b","role":"sequence","name":"veronese_section_2_1 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:  3;\n  1:  5,    8;\n  2:  6,   24;\n  3:  8,   76,    24;\n  4:  9,  126,   108;\n  5: 11,  258,   504,    64;\n  6: 12,  360,  1080,   384;\n  7: 14,  608,  3006,  2368,  160;\n  8: 15,  780,  5049,  6240, 1200;\n  9: 17, 1180, 10959, 22000, 9200, 384;\n","recurrence_sentence":"The row polynomials satisfy 32*P(n,x) = -(706-2280*x+2208*x^2-256*x^3)*P(n-6,x) + (81-549*x+1044*x^2-576*x^3)*d/dx P(n-6,x) - (888-768*x+384*x^2)*P(n-4,x) - (162+36*x-576*x^2)*d/dx P(n-4,x) + (168+192*x)*P(n-2,x) + (81-144*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 3, P(1,x) = 5 + 8*x, P(2,x) = 6 + 24*x, P(3,x) = 8 + 76*x + 24*x^2, P(4,x) = 9 + 126*x + 108*x^2, P(5,x) = 11 + 258*x + 504*x^2 + 64*x^3.","mathematica":"P[0, x_] := 3;\nP[1, x_] := 5 + 8*x;\nP[2, x_] := 6 + 24*x;\nP[3, x_] := 8 + 76*x + 24*x^2;\nP[4, x_] := 9 + 126*x + 108*x^2;\nP[5, x_] := 11 + 258*x + 504*x^2 + 64*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(706 - 2280*x + 2208*x^2 - 256*x^3)*P[n - 6, x] + (81 - 549*x + 1044*x^2 - 576*x^3)*D[P[n - 6, x], x] - (888 - 768*x + 384*x^2)*P[n - 4, x] - (162 + 36*x - 576*x^2)*D[P[n - 4, x], x] + (168 + 192*x)*P[n - 2, x] + (81 - 144*x)*D[P[n - 2, x], x])/(32)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 5 + 8*x; v[3] = 6 + 24*x; v[4] = 8 + 76*x + 24*x^2; v[5] = 9 + 126*x + 108*x^2; v[6] = 11 + 258*x + 504*x^2 + 64*x^3; for(n = 6, N, my(i = n + 1); v[i] = (-(706 - 2280*x + 2208*x^2 - 256*x^3)*v[i - 6] + (81 - 549*x + 1044*x^2 - 576*x^3)*deriv(v[i - 6], x) - (888 - 768*x + 384*x^2)*v[i - 4] - (162 + 36*x - 576*x^2)*deriv(v[i - 4], x) + (168 + 192*x)*v[i - 2] + (81 - 144*x)*deriv(v[i - 2], x))/(32)); 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-5a70a6e97131d437","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-1f83abdcb5264ea5","e-cca56a90043fa45b"],"text":"A064861 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A064861 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}