{"id":"n-43db9268f0694442","revision":0,"first_index":1,"fingerprint":"f6-b05b95bae6fcd71d","name":"theta_plus_one_squared of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","4"],["0","4","9"],["0","0","27","16"],["0","0","9","96","25"],["0","0","0","80","250","36"],["0","0","0","16","375","540","49"],["0","0","0","0","175","1260","1029","64"],["0","0","0","0","25","1008","3430","1792","81"],["0","0","0","0","0","324","4116","8064","2916","100"],["0","0","0","0","0","36","2205","13440","17010","4500","121"],["0","0","0","0","0","0","539","10560","37422","33000","6655","144"],["0","0","0","0","0","0","49","4224","40095","92400","59895","9504","169"],["0","0","0","0","0","0","0","832","23166","128700","207636","102960","13182","196"],["0","0","0","0","0","0","0","64","7371","100100","363363","432432","169169","17836","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","13","43","130","366","980","2528","6336","15520","37312","88320","206336","476672","1090560"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","4"]],"recurrence":{"denominator":{"coeffs":[["0","24","-72"],["0","40","276"],["-2","-102","-416"],["-3","30","308"],["0","20","-112"],["1","-8","16"]]},"terms":[{"coeff":{"coeffs":[["0","-8","-8","16"],["0","-16","-200","-120"],["2","0","-70","0"],["3","-39","126","252"],["0","0","60","-208"],["-1","11","-40","48"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","8","-16","8"],["0","0","28","28","-56"],["0","2","42","-52","8"],["0","4","-20","-48","64"],["0","2","-18","48","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171841,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-0f4e3c87584b8827"]},{"id":171842,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0f4e3c87584b8827"]},{"id":171843,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0f4e3c87584b8827"]}],"region_exact":{},"incoming_edges":[{"id":"e-0f4e3c87584b8827","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-43db9268f0694442","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":"7f5c7a01b53847b9c552897547c9cdf2aaba8eb06d27e52353b0078e74860275","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cf23198864b5678a","role":"sequence","name":"Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 4;\n   3: 0, 4,  9;\n   4: 0, 0, 27, 16;\n   5: 0, 0,  9, 96,  25;\n   6: 0, 0,  0, 80, 250,   36;\n   7: 0, 0,  0, 16, 375,  540,   49;\n   8: 0, 0,  0,  0, 175, 1260, 1029,   64;\n   9: 0, 0,  0,  0,  25, 1008, 3430, 1792,   81;\n  10: 0, 0,  0,  0,   0,  324, 4116, 8064, 2916, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^5-3*n^3-2*n^2-(8*n^5-20*n^4-30*n^3+102*n^2-40*n-24)*x+(16*n^5-112*n^4+308*n^3-416*n^2+276*n-72)*x^2)*P(n,x) = -(n^5-3*n^3-2*n^2-(11*n^5-39*n^3-16*n-8)*x+(40*n^5-60*n^4-126*n^3+70*n^2+200*n+8)*x^2-(48*n^5-208*n^4+252*n^3-120*n+16)*x^3)*P(n-1,x) + ((2*n^4+4*n^3+2*n^2)*x-(18*n^4+20*n^3-42*n^2-28*n-8)*x^2+(48*n^4-48*n^3-52*n^2+28*n-16)*x^3-(32*n^4-64*n^3-8*n^2+56*n-8)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 3*n^3 - 2*n^2 - (11*n^5 - 39*n^3 - 16*n - 8)*x + (40*n^5 - 60*n^4 - 126*n^3 + 70*n^2 + 200*n + 8)*x^2 - (48*n^5 - 208*n^4 + 252*n^3 - 120*n + 16)*x^3)*P[n - 1, x] + ((2*n^4 + 4*n^3 + 2*n^2)*x - (18*n^4 + 20*n^3 - 42*n^2 - 28*n - 8)*x^2 + (48*n^4 - 48*n^3 - 52*n^2 + 28*n - 16)*x^3 - (32*n^4 - 64*n^3 - 8*n^2 + 56*n - 8)*x^4)*D[P[n - 1, x], x])/(n^5 - 3*n^3 - 2*n^2 - (8*n^5 - 20*n^4 - 30*n^3 + 102*n^2 - 40*n - 24)*x + (16*n^5 - 112*n^4 + 308*n^3 - 416*n^2 + 276*n - 72)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; for(n = 3, N, my(i = n); v[i] = (-(n^5 - 3*n^3 - 2*n^2 - (11*n^5 - 39*n^3 - 16*n - 8)*x + (40*n^5 - 60*n^4 - 126*n^3 + 70*n^2 + 200*n + 8)*x^2 - (48*n^5 - 208*n^4 + 252*n^3 - 120*n + 16)*x^3)*v[i - 1] + ((2*n^4 + 4*n^3 + 2*n^2)*x - (18*n^4 + 20*n^3 - 42*n^2 - 28*n - 8)*x^2 + (48*n^4 - 48*n^3 - 52*n^2 + 28*n - 16)*x^3 - (32*n^4 - 64*n^3 - 8*n^2 + 56*n - 8)*x^4)*deriv(v[i - 1], x))/(n^5 - 3*n^3 - 2*n^2 - (8*n^5 - 20*n^4 - 30*n^3 + 102*n^2 - 40*n - 24)*x + (16*n^5 - 112*n^4 + 308*n^3 - 416*n^2 + 276*n - 72)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cf23198864b5678a","steps":["(\u03b8+1)\u00b2"],"edges":["e-0f4e3c87584b8827"],"text":"A098158 \u2192 (\u03b8+1)\u00b2"},"headline":"A098158 \u2192 (\u03b8+1)\u00b2","chain":null}