{"id":"n-3e5a2dffd536ebb5","revision":0,"first_index":0,"fingerprint":"f6-d3e13aaaafda97c1","name":"one_plus_derivative of derivative of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["2"],["12","6"],["48","60","12"],["140","360","180","20"],["330","1500","1500","420","30"],["672","4830","8400","4620","840","42"],["1232","12936","35280","33320","11760","1512","56"],["2088","30240","119952","176400","105840","26208","2520","72"],["3330","63720","347760","740880","687960","287280","52920","3960","90"],["5060","123750","891000","2605680","3492720","2245320","693000","99000","5940","110"],["7392","225060","2069100","7971480","14636160","13538448","6403320","1524600","174240","8580","132"],["10452","387816","4435860","21801780","52664040","66594528","45189144","16422120","3114540","291720","12012","156"],["14378","638820","8900892","54394340","167477310","278269992","257657400","133981848","38648610","5985980","468468","16380","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","18","120","700","3780","19404","96096","463320","2187900","10161580","46558512","210924168","946454600"],"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"],["2"]],"recurrence":{"denominator":{"coeffs":[["-16","14","-2"],["18","-11","1"],["-7","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-30","36","-6"],["4","34","-29","3"],["-3","-17","6","0"],["1","3","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-18","12","-2"],["0","-7","11","-4","0"],["0","2","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175071,"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-26d17d88bd48440e"]},{"id":175072,"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-26d17d88bd48440e"]},{"id":175073,"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-26d17d88bd48440e"]}],"region_exact":{},"incoming_edges":[{"id":"e-26d17d88bd48440e","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-3e5a2dffd536ebb5","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":"42b18fa03abee577bc6df21fb8e665a8e7e072f51ba7105ee7228d54b483e826","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-4ea7eb31fd1a1554","role":"sequence","name":"derivative of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","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-1d46b0fc1dc776b2","kind":"fingerprint_related","node_a_id":"n-15fe5ebf4c0a4758","node_b_id":"n-3e5a2dffd536ebb5","fingerprint":"f6-d3e13aaaafda97c1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d3e13aaaafda97c1"},"other_id":"n-15fe5ebf4c0a4758","other":{"id":"n-15fe5ebf4c0a4758","name":"one_plus_derivative of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-d3e13aaaafda97c1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","5","1"]],"card_rows":[["1"],["2","1"],["4","5","1"],["7","18","9","1"]],"row_sums_preview":["1","3","10","35","126","462","1716","6435"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["D+1"],"edges":["e-56b10b759eb929ba"],"text":"A001263 \u2192 D+1"},"headline":"A001263 \u2192 D+1","chain":null}},{"id":"l-55f483c87e1916ad","kind":"fingerprint_related","node_a_id":"n-085cc2fc18757071","node_b_id":"n-3e5a2dffd536ebb5","fingerprint":"f6-d3e13aaaafda97c1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d3e13aaaafda97c1"},"other_id":"n-085cc2fc18757071","other":{"id":"n-085cc2fc18757071","name":"one_plus_derivative of narayana","first_index":1,"fingerprint":"f6-d3e13aaaafda97c1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","5","1"]],"card_rows":[["1"],["2","1"],["4","5","1"],["7","18","9","1"]],"row_sums_preview":["1","3","10","35","126","462","1716","6435"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["D+1"],"edges":["e-c4aee2e8ba50f587"],"text":"narayana \u2192 D+1"},"headline":"narayana \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:    2;\n  3:   12,     6;\n  4:   48,    60,     12;\n  5:  140,   360,    180,     20;\n  6:  330,  1500,   1500,    420,     30;\n  7:  672,  4830,   8400,   4620,    840,    42;\n  8: 1232, 12936,  35280,  33320,  11760,  1512,   56;\n  9: 2088, 30240, 119952, 176400, 105840, 26208, 2520, 72;\n","recurrence_sentence":"The row polynomials satisfy (n^3-7*n^2+18*n-16+(2*n^2-11*n+14)*x+(n-2)*x^2)*P(n,x) = (n^3-3*n^2+4*n+(3*n^3-17*n^2+34*n-30)*x+(6*n^2-29*n+36)*x^2+(3*n-6)*x^3)*P(n-1,x) + ((2*n^2-7*n+8)*x-(2*n^2-11*n+18)*x^2-(4*n-12)*x^3-2*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n^2 + 4*n + (3*n^3 - 17*n^2 + 34*n - 30)*x + (6*n^2 - 29*n + 36)*x^2 + (3*n - 6)*x^3)*P[n - 1, x] + ((2*n^2 - 7*n + 8)*x - (2*n^2 - 11*n + 18)*x^2 - (4*n - 12)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 7*n^2 + 18*n - 16 + (2*n^2 - 11*n + 14)*x + (n - 2)*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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 3*n^2 + 4*n + (3*n^3 - 17*n^2 + 34*n - 30)*x + (6*n^2 - 29*n + 36)*x^2 + (3*n - 6)*x^3)*v[i - 1] + ((2*n^2 - 7*n + 8)*x - (2*n^2 - 11*n + 18)*x^2 - (4*n - 12)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 7*n^2 + 18*n - 16 + (2*n^2 - 11*n + 14)*x + (n - 2)*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-b1b19325e0094118","steps":["D","D+1"],"edges":["e-2f63263d248c26b4","e-26d17d88bd48440e"],"text":"A367270 \u2192 D \u2192 D+1"},"headline":"A367270 \u2192 D \u2192 D+1","chain":null}