{"id":"n-62c8c2cd2df5630e","revision":0,"first_index":1,"fingerprint":"f6-7b9a7ea9d38360c0","name":"theta_plus_one of one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 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":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["2","8","3"],["2","20","21","4"],["2","36","78","40","5"],["2","52","198","200","65","6"],["2","68","390","664","410","96","7"],["2","84","654","1672","1690","732","133","8"],["2","100","990","3480","5210","3612","1190","176","9"],["2","116","1398","6344","13050","13212","6846","1808","225","10"],["2","132","1878","10520","28090","38844","29022","11888","2610","280","11"],["2","148","2430","16264","54010","96924","98182","57360","19314","3620","341","12"],["2","164","3054","23832","95290","213276","279510","219760","104562","29780","4862","408","13"],["2","180","3750","33480","157210","425436","695310","704272","447858","178820","44022","6360","481","14"],["2","196","4518","45464","245850","784956","1553678","1959024","1597554","847220","290422","62856","8138","560","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","47","161","523","1637","4975","14769","43011","123277","348607","974513","2697195","7400453"],"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","2"],["2","8","3"],["2","20","21","4"],["2","36","78","40","5"],["2","52","198","200","65","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","0","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","0","3","3"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["0","3","3","3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1","-3","-3","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3","-3","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174028,"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-f6e090215616d45c"]},{"id":174029,"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-f6e090215616d45c"]},{"id":174030,"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-f6e090215616d45c"]}],"region_exact":{},"incoming_edges":[{"id":"e-f6e090215616d45c","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-62c8c2cd2df5630e","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":"512090dd4143066eb8200cdbdddeb8b334dc5d251cd6a1b831cf3d9b4043d776","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-6115e40a0e752de9","role":"sequence","name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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,   2;\n   3: 2,   8,    3;\n   4: 2,  20,   21,    4;\n   5: 2,  36,   78,   40,     5;\n   6: 2,  52,  198,  200,    65,     6;\n   7: 2,  68,  390,  664,   410,    96,    7;\n   8: 2,  84,  654, 1672,  1690,   732,  133,    8;\n   9: 2, 100,  990, 3480,  5210,  3612, 1190,  176,   9;\n  10: 2, 116, 1398, 6344, 13050, 13212, 6846, 1808, 225, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x^3*P(n-6,x) + (3*x^2+3*x^3)*P(n-5,x) + (3*x+3*x^2+3*x^3)*P(n-4,x) + (1-3*x-3*x^2+x^3)*P(n-3,x) - (3+3*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 2 + 8*x + 3*x^2, P(4,x) = 2 + 20*x + 21*x^2 + 4*x^3, P(5,x) = 2 + 36*x + 78*x^2 + 40*x^3 + 5*x^4, P(6,x) = 2 + 52*x + 198*x^2 + 200*x^3 + 65*x^4 + 6*x^5.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[3, x_] := 2 + 8*x + 3*x^2;\nP[4, x_] := 2 + 20*x + 21*x^2 + 4*x^3;\nP[5, x_] := 2 + 36*x + 78*x^2 + 40*x^3 + 5*x^4;\nP[6, x_] := 2 + 52*x + 198*x^2 + 200*x^3 + 65*x^4 + 6*x^5;\nP[n_, x_] := P[n, x] = Expand[x^3*P[n - 6, x] + (3*x^2 + 3*x^3)*P[n - 5, x] + (3*x + 3*x^2 + 3*x^3)*P[n - 4, x] + (1 - 3*x - 3*x^2 + x^3)*P[n - 3, x] - (3 + 3*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 2 + 8*x + 3*x^2; v[4] = 2 + 20*x + 21*x^2 + 4*x^3; v[5] = 2 + 36*x + 78*x^2 + 40*x^3 + 5*x^4; v[6] = 2 + 52*x + 198*x^2 + 200*x^3 + 65*x^4 + 6*x^5; for(n = 7, N, my(i = n); v[i] = x^3*v[i - 6] + (3*x^2 + 3*x^3)*v[i - 5] + (3*x + 3*x^2 + 3*x^3)*v[i - 4] + (1 - 3*x - 3*x^2 + x^3)*v[i - 3] - (3 + 3*x + 3*x^2)*v[i - 2] + (3 + 3*x)*v[i - 1]); 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-b0fc620949fb5990","steps":["D+1","\u03b8+1"],"edges":["e-87cb78b62855bfe8","e-f6e090215616d45c"],"text":"A122542 \u2192 D+1 \u2192 \u03b8+1"},"headline":"A122542 \u2192 D+1 \u2192 \u03b8+1","chain":null}