{"id":"n-e5037b63434d4540","revision":0,"first_index":0,"fingerprint":"f6-3ff6c47fdabd4f1c","name":"derivative of basis_map_A007318 of Delannoy paths counted by number of weak peaks.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["16","8"],["114","120","24"],["768","1248","576","64"],["5010","11040","8160","2240","160"],["32016","89160","91200","41280","7680","384"],["201698","679224","889560","566720","178080","24192","896"],["1257472","4968320","7940352","6549760","2938880","688128","71680","2048"],["7777314","35270784","66584448","67560192","39634560","13418496","2451456","202752","4608"],["47800080","244712520","533018880","642673920","467066880","208535040","55695360","8202240","552960","10240"],["292292946","1667290680","4116519000","5753552640","4995661440","2780692992","986979840","214579200","26104320","1464320","22528"],["1779856128","11193598944","30896585280","49133367360","49663088640","33188069376","14770280448","4297328640","778567680","79749120","3784704","49152"],["10799942322","74239495200","226556190432","404016550080","466237268640","364116639744","195420825600","71664113664","17485079040","2688491520","235462656","9584640","106496"],["65336473104","487357505544","1629431189568","3220882496832","4179953057280","3738771324288","2353395652608","1045026318336","322919460864","67251824640","8905408512","675397632","23855104","229376"],["394246725570","3171484988280","11529221460120","25022591455680","36081078153120","36388778025600","26310517833600","13747746355200","5163468710400","1367945779200","246651525120","28469575680","1889771520","58490880","491520"],["2373580947456","20483318545920","80443843200000","190189542466560","301723906498560","338854434766848","276929299046400","166667233075200","73886072832000","23875679027200","5498720747520","868445061120","88263884800","5174722560","141557760","1048576"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["24","258","2656","26610","261720","2540370","24416640","232904610","2208268120","20835159906","195784325472","1833469750034","17119930193688","159454149341490","1481887657428480"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["16","8"]],"recurrence":{"denominator":{"coeffs":[["1"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","4"],["16","12"],["7","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-9","-6"],["-2","-6","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187634,"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-a29a4e3444e6e58b"]},{"id":187635,"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-a29a4e3444e6e58b"]},{"id":187636,"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-a29a4e3444e6e58b"]}],"region_exact":{},"incoming_edges":[{"id":"e-a29a4e3444e6e58b","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-e5037b63434d4540","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":"f38e3a9193c36a3da8a9bb4fd25483082dc3017daf890f31543e696654d845ff","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-695f6bafb476fd66","role":"sequence","name":"basis_map_A007318 of Delannoy paths counted by number of weak peaks.","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-52a554e3d48811c2","kind":"fingerprint_related","node_a_id":"n-cde9ae93bdb6208d","node_b_id":"n-e5037b63434d4540","fingerprint":"f6-3ff6c47fdabd4f1c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3ff6c47fdabd4f1c"},"other_id":"n-cde9ae93bdb6208d","other":{"id":"n-cde9ae93bdb6208d","name":"mobius_x_over_x_plus_one of theta_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","first_index":1,"fingerprint":"f6-3ff6c47fdabd4f1c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["4","8"],["8","40","38"]],"card_rows":[["2"],["4","8"],["8","40","38"],["16","144","312","192"]],"row_sums_preview":["2","12","86","664","5322","43620","362910","3052080"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-d14d9d630db1a5b2","e-966317e932a7cbd2"],"text":"A114655 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A114655 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7ec68dfe9b8ac7ca","kind":"fingerprint_related","node_a_id":"n-b455bb8d8c0e1d5c","node_b_id":"n-e5037b63434d4540","fingerprint":"f6-3ff6c47fdabd4f1c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3ff6c47fdabd4f1c"},"other_id":"n-b455bb8d8c0e1d5c","other":{"id":"n-b455bb8d8c0e1d5c","name":"mobius_x_over_x_plus_one of theta_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","first_index":1,"fingerprint":"f6-3ff6c47fdabd4f1c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["4","20","19"]],"card_rows":[["1"],["2","4"],["4","20","19"],["8","72","156","96"]],"row_sums_preview":["1","6","43","332","2661","21810","181455","1526040"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-57c7b63013beaaa7","e-6ef7ffa1390aca1b"],"text":"A114656 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A114656 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-895fc22675274731","kind":"fingerprint_related","node_a_id":"n-a16d6c9bf5373530","node_b_id":"n-e5037b63434d4540","fingerprint":"f6-3ff6c47fdabd4f1c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3ff6c47fdabd4f1c"},"other_id":"n-a16d6c9bf5373530","other":{"id":"n-a16d6c9bf5373530","name":"derivative of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","first_index":1,"fingerprint":"f6-3ff6c47fdabd4f1c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["16","8"]],"card_rows":[["2"],["16","8"],["114","120","24"],["768","1248","576","64"]],"row_sums_preview":["0","2","24","258","2656","26610","261720","2540370"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A013609","D"],"edges":["e-2f2f885dba2c5644","e-fc1937f935577eac"],"text":"A104684 \u2192 basis A013609 \u2192 D"},"headline":"A104684 \u2192 basis A013609 \u2192 D","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:        16,          8;\n  1:       114,        120,         24;\n  2:       768,       1248,        576,         64;\n  3:      5010,      11040,       8160,       2240,        160;\n  4:     32016,      89160,      91200,      41280,       7680,        384;\n  5:    201698,     679224,     889560,     566720,     178080,      24192,       896;\n  6:   1257472,    4968320,    7940352,    6549760,    2938880,     688128,     71680,      2048;\n  7:   7777314,   35270784,   66584448,   67560192,   39634560,   13418496,   2451456,    202752,     4608;\n  8:  47800080,  244712520,  533018880,  642673920,  467066880,  208535040,  55695360,   8202240,   552960,   10240;\n  9: 292292946, 1667290680, 4116519000, 5753552640, 4995661440, 2780692992, 986979840, 214579200, 26104320, 1464320, 22528;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+1)*P(n,x) = (7*n^2+16*n+8+(6*n^2+12*n+4)*x)*P(n-1,x) - (2*n+3+(6*n+9)*x+(4*n+6)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 16 + 8*x.","mathematica":"P[0, x_] := 16 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n^2 + 16*n + 8 + (6*n^2 + 12*n + 4)*x)*P[n - 1, x] - (2*n + 3 + (6*n + 9)*x + (4*n + 6)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 16 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^2 + 16*n + 8 + (6*n^2 + 12*n + 4)*x)*v[i - 1] - (2*n + 3 + (6*n + 9)*x + (4*n + 6)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*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-b31aa1a3fd93d174","steps":["basis A007318","D"],"edges":["e-9a0dabc7f010818b","e-a29a4e3444e6e58b"],"text":"A133214 \u2192 basis A007318 \u2192 D"},"headline":"A133214 \u2192 basis A007318 \u2192 D","chain":null}