{"id":"n-430b08780a40ace0","revision":0,"first_index":0,"fingerprint":"f6-3dfa0014f17e9a3d","name":"one_plus_derivative 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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["2"],["5","2"],["10","6"],["17","18","3"],["26","44","12"],["37","90","42","4"],["50","162","120","20"],["65","266","285","80","5"],["82","408","588","260","30"],["101","594","1092","700","135","6"],["122","830","1872","1624","490","42"],["145","1122","3015","3360","1470","210","7"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","2","7","16","38","82","173","352","701","1368","2628","4980","9329"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-20","34","-8"],["22","-25","4"],["-8","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["-10","-37","74","-24"],["13","33","-46","8"],["-6","-14","8","0"],["1","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-5","-11","32","-16"],["0","4","12","-16","0"],["0","-1","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171180,"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-131fa8a70cb41a31"]},{"id":171181,"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-131fa8a70cb41a31"]},{"id":171182,"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-131fa8a70cb41a31"]}],"region_exact":{},"incoming_edges":[{"id":"e-131fa8a70cb41a31","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-430b08780a40ace0","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"5bfc34f4186f109169e0378b4b5387d1b4edeec94fc3ecec46fd717212ee7935","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-d30abaa0df7d695e","role":"sequence","name":"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.","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-cf8dfa0154d7792b","kind":"fingerprint_related","node_a_id":"n-430b08780a40ace0","node_b_id":"n-ca94fd13062d3f7a","fingerprint":"f6-3dfa0014f17e9a3d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3dfa0014f17e9a3d"},"other_id":"n-ca94fd13062d3f7a","other":{"id":"n-ca94fd13062d3f7a","name":"one_plus_derivative of derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","first_index":0,"fingerprint":"f6-3dfa0014f17e9a3d","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["5","2"],["10","6"]],"row_sums_preview":["0","1","2","7","16","38","82","173"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-569a801b809b6abd","steps":["D","D+1"],"edges":["e-45a7ac4082b2eab9","e-c2bd9cc39f98a9f6"],"text":"A169803 \u2192 D \u2192 D+1"},"headline":"A169803 \u2192 D \u2192 D+1","chain":null}},{"id":"l-e83a3a94cc57df86","kind":"fingerprint_related","node_a_id":"n-08dd159ee1b67a3f","node_b_id":"n-430b08780a40ace0","fingerprint":"f6-3dfa0014f17e9a3d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3dfa0014f17e9a3d"},"other_id":"n-08dd159ee1b67a3f","other":{"id":"n-08dd159ee1b67a3f","name":"one_plus_derivative 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":"f6-3dfa0014f17e9a3d","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"],["5","2"],["10","6"]],"row_sums_preview":["0","0","1","2","7","16","38","82"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["D","D+1"],"edges":["e-076e29b3e57f3252","e-d4580406aaef86c1"],"text":"A052553 \u2192 D \u2192 D+1"},"headline":"A052553 \u2192 D \u2192 D+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:  0;\n  2:  1;\n  3:  2;\n  4:  5,   2;\n  5: 10,   6;\n  6: 17,  18,   3;\n  7: 26,  44,  12;\n  8: 37,  90,  42,  4;\n  9: 50, 162, 120, 20;\n","recurrence_sentence":"The row polynomials satisfy (n^3-8*n^2+22*n-20+(4*n^2-25*n+34)*x+(4*n-8)*x^2)*P(n,x) = (n^3-6*n^2+13*n-10+(2*n^3-14*n^2+33*n-37)*x+(8*n^2-46*n+74)*x^2+(8*n-24)*x^3)*P(n-1,x) - ((n^2-4*n+5)*x+(4*n^2-12*n+11)*x^2+(16*n-32)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 6*n^2 + 13*n - 10 + (2*n^3 - 14*n^2 + 33*n - 37)*x + (8*n^2 - 46*n + 74)*x^2 + (8*n - 24)*x^3)*P[n - 1, x] - ((n^2 - 4*n + 5)*x + (4*n^2 - 12*n + 11)*x^2 + (16*n - 32)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - 8*n^2 + 22*n - 20 + (4*n^2 - 25*n + 34)*x + (4*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 6*n^2 + 13*n - 10 + (2*n^3 - 14*n^2 + 33*n - 37)*x + (8*n^2 - 46*n + 74)*x^2 + (8*n - 24)*x^3)*v[i - 1] - ((n^2 - 4*n + 5)*x + (4*n^2 - 12*n + 11)*x^2 + (16*n - 32)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - 8*n^2 + 22*n - 20 + (4*n^2 - 25*n + 34)*x + (4*n - 8)*x^2)); 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-75ccd0453a9b6cec","steps":["D","D+1"],"edges":["e-79fb0bd874276b0d","e-131fa8a70cb41a31"],"text":"A011973 \u2192 D \u2192 D+1"},"headline":"A011973 \u2192 D \u2192 D+1","chain":null}
