{"id":"n-3c0126f4fb6ddb15","revision":1,"first_index":0,"fingerprint":"f6-393ea6ec08e838b3","name":"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","2"],["4","6"],["5","12","3"],["6","20","12"],["7","30","30","4"],["8","42","60","20"],["9","56","105","60","5"],["10","72","168","140","30"],["11","90","252","280","105","6"],["12","110","360","504","280","42"],["13","132","495","840","630","168","7"],["14","156","660","1320","1260","504","56"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","2","5","10","20","38","71","130","235","420","744","1308","2285","3970"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-4"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160885,"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-917662e9096bb187"]},{"id":160886,"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-917662e9096bb187"]},{"id":160887,"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-917662e9096bb187"]}],"region_exact":{},"incoming_edges":[{"id":"e-917662e9096bb187","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-3c0126f4fb6ddb15","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":"ce3dd6806fe6caeb8b18f68ab547e8c8129e5b816f83d89e8533034b04884335","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-569a801b809b6abd","role":"sequence","name":"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-b72550bb303166d9","kind":"fingerprint_related","node_a_id":"n-d30abaa0df7d695e","node_b_id":"n-3c0126f4fb6ddb15","fingerprint":"f6-393ea6ec08e838b3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-393ea6ec08e838b3"},"other_id":"n-d30abaa0df7d695e","other":{"id":"n-d30abaa0df7d695e","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.","first_index":0,"fingerprint":"f6-393ea6ec08e838b3","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","2"],["4","6"]],"row_sums_preview":["0","0","1","2","5","10","20","38"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75ccd0453a9b6cec","steps":["D"],"edges":["e-79fb0bd874276b0d"],"text":"A011973 \u2192 D"},"headline":"A011973 \u2192 D","chain":null}},{"id":"l-b9ab07e8f31fe387","kind":"fingerprint_related","node_a_id":"n-05baa90780a27a2f","node_b_id":"n-3c0126f4fb6ddb15","fingerprint":"f6-393ea6ec08e838b3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-393ea6ec08e838b3"},"other_id":"n-05baa90780a27a2f","other":{"id":"n-05baa90780a27a2f","name":"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":"f6-393ea6ec08e838b3","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","2"],["4","6"]],"row_sums_preview":["0","0","1","2","5","10","20","38"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["D"],"edges":["e-fbf2495e314f62ea"],"text":"A052553 \u2192 D"},"headline":"A052553 \u2192 D","chain":null}},{"id":"l-bfcebfb638c5d4aa","kind":"fingerprint_related","node_a_id":"n-7697b772cd7a5e3f","node_b_id":"n-3c0126f4fb6ddb15","fingerprint":"f6-393ea6ec08e838b3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-393ea6ec08e838b3"},"other_id":"n-7697b772cd7a5e3f","other":{"id":"n-7697b772cd7a5e3f","name":"theta_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-393ea6ec08e838b3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","2","3"]],"card_rows":[["1"],["0","2"],["0","2","3"],["0","0","6","4"]],"row_sums_preview":["1","2","5","10","20","38","71","130"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-392f49b496f79fbf","steps":["\u03b8+1"],"edges":["e-cd00da36b6a89aa7"],"text":"A026729 \u2192 \u03b8+1"},"headline":"A026729 \u2192 \u03b8+1","chain":null}},{"id":"l-2e95eed10e524d65","kind":"oeis_match","node_a_id":"n-3c0126f4fb6ddb15","node_b_id":null,"fingerprint":null,"oeis_id":"A001629","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001629","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001629","relation":"match_row_sums"}],"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,  2;\n  4: 4,  6;\n  5: 5, 12,   3;\n  6: 6, 20,  12;\n  7: 7, 30,  30,  4;\n  8: 8, 42,  60, 20;\n  9: 9, 56, 105, 60, 5;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n+(2*n-4)*x)*P(n-1,x) - (x+4*x^2)*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 - 4)*x)*P[n - 1, x] - (x + 4*x^2)*D[P[n - 1, x], x])/(n - 1)]];\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 - 4)*x)*v[i - 1] - (x + 4*x^2)*deriv(v[i - 1], x))/(n - 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-569a801b809b6abd","steps":["D"],"edges":["e-917662e9096bb187"],"text":"A169803 \u2192 D"},"headline":"A169803 \u2192 D","chain":null}
