{"id":"n-4a1071f2dabe7b56","revision":0,"first_index":0,"fingerprint":"f6-8495a6d14838bc51","name":"theta_plus_one of 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.","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","2"],["5","6"],["8","14","3"],["13","30","12"],["21","60","36","4"],["34","116","93","20"],["55","218","219","72","5"],["89","402","486","216","30"],["144","730","1032","580","125","6"],["233","1310","2121","1444","425","42"],["377","2328","4248","3400","1275","198","7"],["610","4104","8334","7672","3505","750","56"],["987","7186","16074","16736","9030","2478","294","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","5","11","25","55","121","263","569","1223","2617","5575","11833","25031","52793"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["40","38","12","0"],["-10","33","20","2"],["0","-5","11","3"],["0","0","-1","1"]]},"terms":[{"coeff":{"coeffs":[["90","78","-36","-38","-4"],["-30","74","108","21","-2"],["0","-20","15","28","4"],["0","0","-3","1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["25","5","-102","-132","-58","-8"],["0","25","20","-40","-47","-12"],["0","0","5","4","-5","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171781,"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-55fe3fa00b004f7e"]},{"id":171782,"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-55fe3fa00b004f7e"]},{"id":171783,"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-55fe3fa00b004f7e"]}],"region_exact":{},"incoming_edges":[{"id":"e-55fe3fa00b004f7e","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-4a1071f2dabe7b56","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":"3d8d557537bbbd3aca451b0c5a56fff9b9d8295a44dd71dc699a5a7ac1e80f70","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,   2;\n  3:  5,   6;\n  4:  8,  14,   3;\n  5: 13,  30,  12;\n  6: 21,  60,  36,   4;\n  7: 34, 116,  93,  20;\n  8: 55, 218, 219,  72,  5;\n  9: 89, 402, 486, 216, 30;\n","recurrence_sentence":"The row polynomials satisfy -(10*n-40+(5*n^2-33*n-38)*x+(n^3-11*n^2-20*n-12)*x^2-(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(30*n-90+(20*n^2-74*n-78)*x+(3*n^3-15*n^2-108*n+36)*x^2-(n^3+28*n^2+21*n-38)*x^3-(2*n^3+4*n^2-2*n-4)*x^4)*P(n-1,x) + (25+(25*n+5)*x+(5*n^2+20*n-102)*x^2+(4*n^2-40*n-132)*x^3-(5*n^2+47*n+58)*x^4-(4*n^2+12*n+8)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 90 + (20*n^2 - 74*n - 78)*x + (3*n^3 - 15*n^2 - 108*n + 36)*x^2 - (n^3 + 28*n^2 + 21*n - 38)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4)*P[n - 1, x] + (25 + (25*n + 5)*x + (5*n^2 + 20*n - 102)*x^2 + (4*n^2 - 40*n - 132)*x^3 - (5*n^2 + 47*n + 58)*x^4 - (4*n^2 + 12*n + 8)*x^5)*D[P[n - 1, x], x])/(-10*n + 40 - (5*n^2 - 33*n - 38)*x - (n^3 - 11*n^2 - 20*n - 12)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(30*n - 90 + (20*n^2 - 74*n - 78)*x + (3*n^3 - 15*n^2 - 108*n + 36)*x^2 - (n^3 + 28*n^2 + 21*n - 38)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4)*v[i - 1] + (25 + (25*n + 5)*x + (5*n^2 + 20*n - 102)*x^2 + (4*n^2 - 40*n - 132)*x^3 - (5*n^2 + 47*n + 58)*x^4 - (4*n^2 + 12*n + 8)*x^5)*deriv(v[i - 1], x))/(-10*n + 40 - (5*n^2 - 33*n - 38)*x - (n^3 - 11*n^2 - 20*n - 12)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)); 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"],"edges":["e-55fe3fa00b004f7e"],"text":"A209599 \u2192 \u03b8+1"},"headline":"A209599 \u2192 \u03b8+1","chain":null}