{"id":"n-608a1d647f43d406","revision":0,"first_index":0,"fingerprint":"f6-edeea83fed38ed02","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["3","6"],["4","20","12"],["5","44","66","20"],["6","80","210","160","30"],["7","130","513","684","325","42"],["8","196","1068","2136","1780","588","56"],["9","280","1995","5492","6865","3990","980","72"],["10","384","3444","12352","21230","18528","8036","1536","90"],["11","510","5598","25176","56285","67542","44058","14928","2295","110"],["12","660","8676","47568","133100","207636","186340","95136","26028","3300","132"],["13","836","12936","84612","288035","561864","655508","460856","190377","43120","4598","156"],["14","1040","18678","143264","580580","1376304","2007110","1835072","1045044","358160","68486","6240","182"],["15","1274","26247","232804","1103960","3111576","5511331","6298664","4667364","2207920","640211","104988","8281","210"],["16","1540","36036","365352","1999140","6584760","13857396","19230672","17816652","10974600","4398108","1096056","156156","10780","240"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","9","36","135","486","1701","5832","19683","65610","216513","708588","2302911","7440174","23914845","76527504"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["3","6"],["4","20","12"]],"recurrence":{"denominator":{"coeffs":[["4"]]},"terms":[{"coeff":{"coeffs":[["8","6","-6","4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1","-3","-3","-4"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-20","-8","-12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","6","8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["14","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174764,"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-9cdadd7d422178fd"]},{"id":174765,"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-9cdadd7d422178fd"]},{"id":174766,"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-9cdadd7d422178fd"]}],"region_exact":{},"incoming_edges":[{"id":"e-9cdadd7d422178fd","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-608a1d647f43d406","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":"c1414ed69cda5f4f50301f893bf54747aa9e20b2c03b93f2e87183094b83e060","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-539ff84504d2bf3a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= 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:  2;\n  1:  3,   6;\n  2:  4,  20,   12;\n  3:  5,  44,   66,    20;\n  4:  6,  80,  210,   160,    30;\n  5:  7, 130,  513,   684,   325,    42;\n  6:  8, 196, 1068,  2136,  1780,   588,    56;\n  7:  9, 280, 1995,  5492,  6865,  3990,   980,    72;\n  8: 10, 384, 3444, 12352, 21230, 18528,  8036,  1536,   90;\n  9: 11, 510, 5598, 25176, 56285, 67542, 44058, 14928, 2295, 110;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = (8+6*x-6*x^2+4*x^3)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (20+8*x+12*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (14+12*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 3 + 6*x, P(2,x) = 4 + 20*x + 12*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 6*x;\nP[2, x_] := 4 + 20*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + 6*x - 6*x^2 + 4*x^3)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (20 + 8*x + 12*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (14 + 12*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 6*x; v[3] = 4 + 20*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((8 + 6*x - 6*x^2 + 4*x^3)*v[i - 3] + (1 - 3*x - 3*x^2 - 4*x^3)*deriv(v[i - 3], x) - (20 + 8*x + 12*x^2)*v[i - 2] - (2 - 6*x - 8*x^2)*deriv(v[i - 2], x) + (14 + 12*x)*v[i - 1] + (1 - 4*x)*deriv(v[i - 1], x))/(4)); 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-539ff84504d2bf3a","steps":["\u03b8+1"],"edges":["e-9cdadd7d422178fd"],"text":"A330510 \u2192 \u03b8+1"},"headline":"A330510 \u2192 \u03b8+1","chain":null}