{"id":"n-eeaaa106a2fccb04","revision":0,"first_index":1,"fingerprint":"f6-53ae1d6e5761f369","name":"theta_plus_one of veronese_section_2_0 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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":[["1"],["1"],["1","10"],["1","70"],["1","252","27"],["1","660","495"],["1","1430","3861","52"],["1","2730","19305","1820"],["1","4760","72930","24752","85"],["1","7752","226746","201552","4845"],["1","11970","610470","1175720","101745","126"],["1","17710","1470942","5408312","1225785","10626"],["1","25300","3244725","20801200","10214875","318780","175"],["1","35100","6660225","69535440","65189475","5328180","20475"],["1","47502","12876435","207583740","339319575","60090030","831285","232"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","11","71","280","1156","5344","23856","102528","440896","1900032","8133376","34605056","146768896","620748800"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"],["1","10"],["1","70"],["1","252","27"],["1","660","495"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["191","-381","189","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-64","192","-192","64"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["3342","-2252","-66"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["-128","1264","-1120","-16"]]},"deriv_order":1,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["9041","146","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1152","5184","64"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1860","52"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2048","416","32"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-111","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1472","128"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["14"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-384","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176519,"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-eae7b99297c964b1"]},{"id":176520,"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-eae7b99297c964b1"]},{"id":176521,"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-eae7b99297c964b1"]}],"region_exact":{},"incoming_edges":[{"id":"e-eae7b99297c964b1","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-eeaaa106a2fccb04","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. 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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":"6676e6de2a6f44c5604b37e94e1d8fad4e04be385bf48b8223d4e9f9c8c2bf5b","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-6a79351bf6797e1b","role":"sequence","name":"veronese_section_2_0 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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: 1;\n   3: 1,   10;\n   4: 1,   70;\n   5: 1,  252,     27;\n   6: 1,  660,    495;\n   7: 1, 1430,   3861,     52;\n   8: 1, 2730,  19305,   1820;\n   9: 1, 4760,  72930,  24752,   85;\n  10: 1, 7752, 226746, 201552, 4845;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (191-381*x+189*x^2+x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3342-2252*x-66*x^2)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (9041+146*x-3*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) + (1860+52*x)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) - (111-3*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) + 14*P(n-1,x) - (384+16*x)*d/dx P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 1 + 10*x, P(4,x) = 1 + 70*x, P(5,x) = 1 + 252*x + 27*x^2, P(6,x) = 1 + 660*x + 495*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 1 + 10*x;\nP[4, x_] := 1 + 70*x;\nP[5, x_] := 1 + 252*x + 27*x^2;\nP[6, x_] := 1 + 660*x + 495*x^2;\nP[n_, x_] := P[n, x] = Expand[(191 - 381*x + 189*x^2 + x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3342 - 2252*x - 66*x^2)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (9041 + 146*x - 3*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] + (1860 + 52*x)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] - (111 - 3*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] + 14*P[n - 1, x] - (384 + 16*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 1 + 10*x; v[4] = 1 + 70*x; v[5] = 1 + 252*x + 27*x^2; v[6] = 1 + 660*x + 495*x^2; for(n = 7, N, my(i = n); v[i] = (191 - 381*x + 189*x^2 + x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3342 - 2252*x - 66*x^2)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (9041 + 146*x - 3*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) + (1860 + 52*x)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) - (111 - 3*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) + 14*v[i - 1] - (384 + 16*x)*deriv(v[i - 1], x)); 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-ba8de4e02acfb6ca","steps":["Veronese 2","\u03b8+1"],"edges":["e-d03002d6b7efed5a","e-eae7b99297c964b1"],"text":"A091042 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A091042 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}