{"id":"n-8faa874efd7868bc","revision":0,"first_index":0,"fingerprint":"f6-5fca0e1aa4d16710","name":"derivative of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1","4"],["0","2","18","24"],["0","0","3","48","150","120"],["0","0","0","4","100","540","980","560"],["0","0","0","0","5","180","1470","4480","5670","2520"],["0","0","0","0","0","6","294","3360","15120","31500","30492","11088"],["0","0","0","0","0","0","7","448","6804","42000","127050","199584","156156","48048"],["0","0","0","0","0","0","0","8","648","12600","101640","415800","936936","1177176","772200","205920"],["0","0","0","0","0","0","0","0","9","900","21780","221760","1171170","3531528","6306300","6589440","3719430","875160"],["0","0","0","0","0","0","0","0","0","10","1210","35640","446160","2942940","11351340","26906880","39673920","35443980","17551820","3695120"],["0","0","0","0","0","0","0","0","0","0","11","1584","55770","840840","6756750","32288256","97201104","189034560","236949570","184756000","81477396","15519504"],["0","0","0","0","0","0","0","0","0","0","0","12","2028","84084","1501500","14414400","83315232","308756448","758238624","1247103000","1357956600","938929992","373173528","64899744"],["0","0","0","0","0","0","0","0","0","0","0","0","13","2548","122850","2562560","28928900","198486288","884611728","2660486400","5499724230","7824416600","7525666148","4672781568","1690097500","270415600"],["0","0","0","0","0","0","0","0","0","0","0","0","0","14","3150","174720","4207840","55135080","442305864","2327925600","8380532160","21125924820","37628330740","47117214144","40562340000","22850118200","7582037400","1123264800"]],"degrees":[-1,1,3,5,7,9,11,13,15,17,19,21,23,25,27],"row_sums":["0","5","44","321","2184","14325","91860","580097","3622928","22437477","138049020","844881345","5148375192","31258302933","189199514532"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1","4"]],"recurrence":{"denominator":{"coeffs":[["1","2"],["-2","-2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","-6","-14","-8"],["0","6","8","0"],["0","-1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-3","-7","-4"],["0","0","2","2","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178244,"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-5269d9c1c8e74708"]},{"id":178245,"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-5269d9c1c8e74708"]},{"id":178246,"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-5269d9c1c8e74708"]}],"region_exact":{},"incoming_edges":[{"id":"e-5269d9c1c8e74708","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-8faa874efd7868bc","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":"0bda49b2880c7dbda0df44144162a9bb66ffa6e529c1895022f810b8ac3585cb","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-e4d248920a317259","role":"sequence","name":"Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","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: 0;\n  1: 1, 4;\n  2: 0, 2, 18, 24;\n  3: 0, 0,  3, 48, 150, 120;\n  4: 0, 0,  0,  4, 100, 540,  980,  560;\n  5: 0, 0,  0,  0,   5, 180, 1470, 4480,  5670,  2520;\n  6: 0, 0,  0,  0,   0,   6,  294, 3360, 15120, 31500,  30492,  11088;\n  7: 0, 0,  0,  0,   0,   0,    7,  448,  6804, 42000, 127050, 199584,  156156,   48048;\n  8: 0, 0,  0,  0,   0,   0,    0,    8,   648, 12600, 101640, 415800,  936936, 1177176,  772200,  205920;\n  9: 0, 0,  0,  0,   0,   0,    0,    0,     9,   900,  21780, 221760, 1171170, 3531528, 6306300, 6589440, 3719430, 875160;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1-(2*n-2)*x)*P(n,x) = -((n^2-6*n+6)*x-(8*n-14)*x^2+8*x^3)*P(n-1,x) + ((2*n-3)*x^2+(2*n-7)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n^2 - 6*n + 6)*x - (8*n - 14)*x^2 + 8*x^3)*P[n - 1, x] + ((2*n - 3)*x^2 + (2*n - 7)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(n^2 - 2*n + 1 - (2*n - 2)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-((n^2 - 6*n + 6)*x - (8*n - 14)*x^2 + 8*x^3)*v[i - 1] + ((2*n - 3)*x^2 + (2*n - 7)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(n^2 - 2*n + 1 - (2*n - 2)*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-e4d248920a317259","steps":["D"],"edges":["e-5269d9c1c8e74708"],"text":"A109983 \u2192 D"},"headline":"A109983 \u2192 D","chain":null}