{"id":"n-d5d5aa9e7b8aa431","revision":1,"first_index":0,"fingerprint":"f6-36286b84fc436bc0","name":"theta_plus_one_squared of Number triangle T(n,k)=C(n,k)C(2,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","4"],["1","16","9"],["0","12","54","16"],["0","0","54","128","25"],["0","0","0","160","250","36"],["0","0","0","0","375","432","49"],["0","0","0","0","0","756","686","64"],["0","0","0","0","0","0","1372","1024","81"],["0","0","0","0","0","0","0","2304","1458","100"],["0","0","0","0","0","0","0","0","3645","2000","121"],["0","0","0","0","0","0","0","0","0","5500","2662","144"],["0","0","0","0","0","0","0","0","0","0","7986","3456","169"],["0","0","0","0","0","0","0","0","0","0","0","11232","4394","196"],["0","0","0","0","0","0","0","0","0","0","0","0","15379","5488","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","26","82","207","446","856","1506","2477","3862","5766","8306","11611","15822","21092"],"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","4"],["1","16","9"],["0","12","54","16"],["0","0","54","128","25"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","0","0","0","1"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","-5"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","10"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","-10"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160993,"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-af85a1ff14d8b9e9"]},{"id":160994,"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-af85a1ff14d8b9e9"]},{"id":160995,"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-af85a1ff14d8b9e9"]}],"region_exact":{},"incoming_edges":[{"id":"e-af85a1ff14d8b9e9","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-d5d5aa9e7b8aa431","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":"32076298a5aee5138f432d539f7d0945ba4b252eafd32504eae4adfba4abd102","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-1815d3e1c9c57d5b","role":"sequence","name":"Number triangle T(n,k)=C(n,k)C(2,n-k).","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,  4;\n  2: 1, 16,  9;\n  3: 0, 12, 54,  16;\n  4: 0,  0, 54, 128,  25;\n  5: 0,  0,  0, 160, 250,  36;\n  6: 0,  0,  0,   0, 375, 432,   49;\n  7: 0,  0,  0,   0,   0, 756,  686,   64;\n  8: 0,  0,  0,   0,   0,   0, 1372, 1024,   81;\n  9: 0,  0,  0,   0,   0,   0,    0, 2304, 1458, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x^5*P(n-5,x) - 5*x^4*P(n-4,x) + 10*x^3*P(n-3,x) - 10*x^2*P(n-2,x) + 5*x*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 1 + 16*x + 9*x^2, P(3,x) = 12*x + 54*x^2 + 16*x^3, P(4,x) = 54*x^2 + 128*x^3 + 25*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 1 + 16*x + 9*x^2;\nP[3, x_] := 12*x + 54*x^2 + 16*x^3;\nP[4, x_] := 54*x^2 + 128*x^3 + 25*x^4;\nP[n_, x_] := P[n, x] = Expand[x^5*P[n - 5, x] - 5*x^4*P[n - 4, x] + 10*x^3*P[n - 3, x] - 10*x^2*P[n - 2, x] + 5*x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 1 + 16*x + 9*x^2; v[4] = 12*x + 54*x^2 + 16*x^3; v[5] = 54*x^2 + 128*x^3 + 25*x^4; for(n = 5, N, my(i = n + 1); v[i] = x^5*v[i - 5] - 5*x^4*v[i - 4] + 10*x^3*v[i - 3] - 10*x^2*v[i - 2] + 5*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["(1/2)D\u00b2+2D+1","(\u03b8+1)\u00b2"],"edges":["e-0c9044372fd705ed","e-af85a1ff14d8b9e9"],"text":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 (\u03b8+1)\u00b2","chain":null}