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DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","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"],["2"],["3","4"],["5","12"],["8","28","9"],["13","60","36"],["21","120","108","16"],["34","232","279","80"],["55","436","657","288","25"],["89","804","1458","864","150"],["144","1460","3096","2320","625","36"],["233","2620","6363","5776","2125","252"],["377","4656","12744","13600","6375","1188","49"],["610","8208","25002","30688","17525","4500","392"],["987","14372","48222","66944","45150","14868","2058","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","7","17","45","109","265","625","1461","3365","7681","17369","38989","86925","192665"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"denominator":{"coeffs":[["-25","190","-69","2","-8","0"],["0","-75","285","-22","-5","0"],["0","0","-60","125","15","1"],["0","0","0","-15","19","2"],["0","0","0","0","-1","1"]]},"terms":[{"coeff":{"coeffs":[["-25","90","271","246","110","24","4"],["0","-75","90","348","261","90","12"],["0","0","-60","20","146","79","13"],["0","0","0","-15","0","21","6"],["0","0","0","0","-1","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-25","190","106","52","-15","-2"],["0","-75","250","103","31","0"],["0","0","-60","110","44","5"],["0","0","0","-15","17","4"],["0","0","0","0","-1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171784,"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-defa4752a255c383"]},{"id":171785,"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-defa4752a255c383"]},{"id":171786,"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-defa4752a255c383"]}],"region_exact":{},"incoming_edges":[{"id":"e-defa4752a255c383","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-defc18c47aef3671","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":"472b9ed5b468eb83565376a99ad49a9b65c57f6650535be894dc203c50b8437f","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-72cbbb2f152c8f9c","role":"sequence","name":"Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","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:  2;\n  2:  3,   4;\n  3:  5,  12;\n  4:  8,  28,    9;\n  5: 13,  60,   36;\n  6: 21, 120,  108,  16;\n  7: 34, 232,  279,  80;\n  8: 55, 436,  657, 288,  25;\n  9: 89, 804, 1458, 864, 150;\n","recurrence_sentence":"The row polynomials satisfy -(25+(75*n-190)*x+(60*n^2-285*n+69)*x^2+(15*n^3-125*n^2+22*n-2)*x^3+(n^4-19*n^3-15*n^2+5*n+8)*x^4-(n^4+2*n^3+n^2)*x^5)*P(n,x) = -(25+(75*n-90)*x+(60*n^2-90*n-271)*x^2+(15*n^3-20*n^2-348*n-246)*x^3+(n^4-146*n^2-261*n-110)*x^4-(21*n^3+79*n^2+90*n+24)*x^5-(n^4+6*n^3+13*n^2+12*n+4)*x^6)*P(n-2,x) - (25+(75*n-190)*x+(60*n^2-250*n-106)*x^2+(15*n^3-110*n^2-103*n-52)*x^3+(n^4-17*n^3-44*n^2-31*n+15)*x^4-(n^4+4*n^3+5*n^2-2)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(25 + (75*n - 90)*x + (60*n^2 - 90*n - 271)*x^2 + (15*n^3 - 20*n^2 - 348*n - 246)*x^3 + (n^4 - 146*n^2 - 261*n - 110)*x^4 - (21*n^3 + 79*n^2 + 90*n + 24)*x^5 - (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x^6)*P[n - 2, x] - (25 + (75*n - 190)*x + (60*n^2 - 250*n - 106)*x^2 + (15*n^3 - 110*n^2 - 103*n - 52)*x^3 + (n^4 - 17*n^3 - 44*n^2 - 31*n + 15)*x^4 - (n^4 + 4*n^3 + 5*n^2 - 2)*x^5)*P[n - 1, x])/(-25 - (75*n - 190)*x - (60*n^2 - 285*n + 69)*x^2 - (15*n^3 - 125*n^2 + 22*n - 2)*x^3 - (n^4 - 19*n^3 - 15*n^2 + 5*n + 8)*x^4 + (n^4 + 2*n^3 + n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(25 + (75*n - 90)*x + (60*n^2 - 90*n - 271)*x^2 + (15*n^3 - 20*n^2 - 348*n - 246)*x^3 + (n^4 - 146*n^2 - 261*n - 110)*x^4 - (21*n^3 + 79*n^2 + 90*n + 24)*x^5 - (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x^6)*v[i - 2] - (25 + (75*n - 190)*x + (60*n^2 - 250*n - 106)*x^2 + (15*n^3 - 110*n^2 - 103*n - 52)*x^3 + (n^4 - 17*n^3 - 44*n^2 - 31*n + 15)*x^4 - (n^4 + 4*n^3 + 5*n^2 - 2)*x^5)*v[i - 1])/(-25 - (75*n - 190)*x - (60*n^2 - 285*n + 69)*x^2 - (15*n^3 - 125*n^2 + 22*n - 2)*x^3 - (n^4 - 19*n^3 - 15*n^2 + 5*n + 8)*x^4 + (n^4 + 2*n^3 + n^2)*x^5)); 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-72cbbb2f152c8f9c","steps":["(\u03b8+1)\u00b2"],"edges":["e-defa4752a255c383"],"text":"A209599 \u2192 (\u03b8+1)\u00b2"},"headline":"A209599 \u2192 (\u03b8+1)\u00b2","chain":null}