{"id":"n-dfae0097fdfbe5b8","revision":0,"first_index":0,"fingerprint":"f6-56d80b07a9b89537","name":"theta_plus_one of veronese_section_2_1 of Triangle of coefficients in expansion of (1+3*x)^n.","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"],["6"],["9","54"],["12","216"],["15","540","729"],["18","1080","4374"],["21","1890","15309","8748"],["24","3024","40824","69984"],["27","4536","91854","314928","98415"],["30","6480","183708","1049760","984150"],["33","8910","336798","2886840","5412825","1062882"],["36","11880","577368","6928416","21651300","12754584"],["39","15444","938223","15011568","70366725","82904796","11160261"],["42","19656","1459458","30023136","197026830","386889048","156243654"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","3","6","63","228","1284","5472","25968","113856","509760","2224128","9708288","41923584","180397056","771661824"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["2","0"],["-2","9"],["0","-18"],["0","9"]]},"terms":[{"coeff":{"coeffs":[["0","72","0"],["-2","-36","-162"],["0","-9","-81"],["0","9","81"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","36","0"],["0","0","18","-162"],["0","0","18","-162"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176180,"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-29d34656b259540a"]},{"id":176181,"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-29d34656b259540a"]},{"id":176182,"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-29d34656b259540a"]}],"region_exact":{},"incoming_edges":[{"id":"e-29d34656b259540a","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-dfae0097fdfbe5b8","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":"9343deff8a3e1146199e52195177929e6d10b695c5ec45c5ce339763b98c0021","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-15cf3f3457596bec","role":"sequence","name":"veronese_section_2_1 of Triangle of coefficients in expansion of (1+3*x)^n.","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:  0;\n  1:  3;\n  2:  6;\n  3:  9,   54;\n  4: 12,  216;\n  5: 15,  540,   729;\n  6: 18, 1080,  4374;\n  7: 21, 1890, 15309,   8748;\n  8: 24, 3024, 40824,  69984;\n  9: 27, 4536, 91854, 314928, 98415;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-2-(9*n^3-18*n^2+9*n)*x)*P(n,x) = -(2*n-(9*n^3-9*n^2-36*n+72)*x-(81*n^3-81*n^2-162*n)*x^2)*P(n-1,x) - (4*x-(18*n^2+18*n+36)*x^2+(162*n^2+162*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - (9*n^3 - 9*n^2 - 36*n + 72)*x - (81*n^3 - 81*n^2 - 162*n)*x^2)*P[n - 1, x] - (4*x - (18*n^2 + 18*n + 36)*x^2 + (162*n^2 + 162*n)*x^3)*D[P[n - 1, x], x])/(-2*n + 2 + (9*n^3 - 18*n^2 + 9*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - (9*n^3 - 9*n^2 - 36*n + 72)*x - (81*n^3 - 81*n^2 - 162*n)*x^2)*v[i - 1] - (4*x - (18*n^2 + 18*n + 36)*x^2 + (162*n^2 + 162*n)*x^3)*deriv(v[i - 1], x))/(-2*n + 2 + (9*n^3 - 18*n^2 + 9*n)*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-6a5af5e1168bb679","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-a72a0a226c85ec5c","e-29d34656b259540a"],"text":"A013610 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A013610 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}