{"id":"n-2da8bd4e7bbc0ddc","revision":0,"first_index":0,"fingerprint":"f6-5bc743e1d3ec7e23","name":"theta_plus_one of derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","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"],["2"],["3","4"],["4","12"],["5","24","9"],["6","40","36"],["7","60","90","16"],["8","84","180","80"],["9","112","315","240","25"],["10","144","504","560","150"],["11","180","756","1120","525","36"],["12","220","1080","2016","1400","252"],["13","264","1485","3360","3150","1008","49"],["14","312","1980","5280","6300","3024","392"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","2","7","16","38","82","173","352","701","1368","2628","4980","9329","17302"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1","0"],["0","1"],["1","-2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-8","0"],["1","-4","-4"],["1","3","-2"],["0","1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4","0"],["0","-1","-5","-4"],["0","0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171343,"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-12a686e60de252ad"]},{"id":171344,"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-12a686e60de252ad"]},{"id":171345,"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-12a686e60de252ad"]}],"region_exact":{},"incoming_edges":[{"id":"e-12a686e60de252ad","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-2da8bd4e7bbc0ddc","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":"c8a539059cb3deac0ab4607573fe572f444620e285acfce4e2a7b79d0767a94d","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-3c0126f4fb6ddb15","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","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":[{"id":"l-69cc10c135653be8","kind":"fingerprint_related","node_a_id":"n-d0df20a156a5cf58","node_b_id":"n-2da8bd4e7bbc0ddc","fingerprint":"f6-5bc743e1d3ec7e23","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5bc743e1d3ec7e23"},"other_id":"n-d0df20a156a5cf58","other":{"id":"n-d0df20a156a5cf58","name":"theta_plus_one of derivative of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","first_index":0,"fingerprint":"f6-5bc743e1d3ec7e23","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["2"],["3","4"],["4","12"]],"row_sums_preview":["0","0","1","2","7","16","38","82"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["D","\u03b8+1"],"edges":["e-79fb0bd874276b0d","e-47ef797c1e85d642"],"text":"A011973 \u2192 D \u2192 \u03b8+1"},"headline":"A011973 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-8e9a4453df3e241f","kind":"fingerprint_related","node_a_id":"n-dd98275404c59d4b","node_b_id":"n-2da8bd4e7bbc0ddc","fingerprint":"f6-5bc743e1d3ec7e23","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5bc743e1d3ec7e23"},"other_id":"n-dd98275404c59d4b","other":{"id":"n-dd98275404c59d4b","name":"theta_plus_one of derivative of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-5bc743e1d3ec7e23","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["2"],["3","4"],["4","12"]],"row_sums_preview":["0","0","1","2","7","16","38","82"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["D","\u03b8+1"],"edges":["e-076e29b3e57f3252","e-d9b2a7bc5d591739"],"text":"A052553 \u2192 D \u2192 \u03b8+1"},"headline":"A052553 \u2192 D \u2192 \u03b8+1","chain":null}}],"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: 0;\n  1: 1;\n  2: 2;\n  3: 3,   4;\n  4: 4,  12;\n  5: 5,  24,   9;\n  6: 6,  40,  36;\n  7: 7,  60,  90,  16;\n  8: 8,  84, 180,  80;\n  9: 9, 112, 315, 240, 25;\n","recurrence_sentence":"The row polynomials satisfy (n^2-1+(n^3-2*n^2+n)*x)*P(n,x) = (n^2+n+(n^3+3*n^2-4*n-8)*x+(2*n^3-2*n^2-4*n)*x^2)*P(n-1,x) - ((n+1)*x+(n^2+5*n+4)*x^2+(4*n^2+4*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (n^3 + 3*n^2 - 4*n - 8)*x + (2*n^3 - 2*n^2 - 4*n)*x^2)*P[n - 1, x] - ((n + 1)*x + (n^2 + 5*n + 4)*x^2 + (4*n^2 + 4*n)*x^3)*D[P[n - 1, x], x])/(n^2 - 1 + (n^3 - 2*n^2 + n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n + (n^3 + 3*n^2 - 4*n - 8)*x + (2*n^3 - 2*n^2 - 4*n)*x^2)*v[i - 1] - ((n + 1)*x + (n^2 + 5*n + 4)*x^2 + (4*n^2 + 4*n)*x^3)*deriv(v[i - 1], x))/(n^2 - 1 + (n^3 - 2*n^2 + n)*x)); 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-569a801b809b6abd","steps":["D","\u03b8+1"],"edges":["e-45a7ac4082b2eab9","e-12a686e60de252ad"],"text":"A169803 \u2192 D \u2192 \u03b8+1"},"headline":"A169803 \u2192 D \u2192 \u03b8+1","chain":null}