{"id":"n-d4ba698c2649182f","revision":0,"first_index":1,"fingerprint":"f6-f603151ffaefdb02","name":"one_plus_derivative of basis_map_A013610 of narayana","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":[["1"],["5","3"],["20","33","9"],["77","225","162","27"],["294","1260","1638","702","81"],["1122","6390","12420","9720","2835","243"],["4290","30591","80190","95175","51030","10935","729"],["16445","141141","468468","758835","623700","246645","40824","2187"],["63206","634920","2558556","5297292","5999994","3667356","1122660","148716","6561"],["243542","2804490","13318344","33729696","49223538","41793570","19901700","4881384","531441","19683"],["940576","12219090","66890070","200849220","360884160","397503288","264692610","101498670","20470320","1869885","59049"],["3640210","52676778","326804940","1136782350","2434728780","3320828280","2885685264","1556006760","492567075","83357505","6495390","177147"],["14115100","225187848","1562140404","6183712260","15411315150","25176131640","27227520144","19257532272","8612887140","2295037800","331264890","22320522","531441"],["54826020","956112300","7335988920","32584307184","92762015346","177046224498","230314982040","203646931488","120006227484","45361205604","10335464568","1289630160","75996063","1594323"],["213286590","4036732875","33950678580","167289230745","536092166610","1172776908303","1788981724530","1910099120787","1412800587198","706400293599","229102797924","45217657485","4932835362","256686003","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","62","491","3975","32730","272940","2298245","19499261","166447388","1427876938","12299750479","106319696611","921771505998","8012155489560"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["4","-11","3"],["-3","-5","3"],["-4","6","0"],["3","0","0"]]},"terms":[{"coeff":{"coeffs":[["-2","13","66","-27"],["12","-2","-114","27"],["-22","-48","54","0"],["12","27","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","7","24","-18"],["0","5","3","-36","0"],["0","-6","-18","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182268,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f9847bed23834472"]},{"id":182269,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-f9847bed23834472"]},{"id":182270,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-f9847bed23834472"]},{"id":182271,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-f9847bed23834472"]}],"region_exact":{},"incoming_edges":[{"id":"e-f9847bed23834472","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-d4ba698c2649182f","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":"0f440534b3b9558ca5cb166276bce2c932d91e390573a2d3fff4910f2cce44b4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9c4b1c1c0d7d0597","role":"sequence","name":"basis_map_A013610 of narayana","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-35fde6e27ec547b3","kind":"fingerprint_related","node_a_id":"n-d072492d6b036063","node_b_id":"n-d4ba698c2649182f","fingerprint":"f6-f603151ffaefdb02","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f603151ffaefdb02"},"other_id":"n-d072492d6b036063","other":{"id":"n-d072492d6b036063","name":"one_plus_derivative of basis_map_A013610 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-f603151ffaefdb02","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","3"],["20","33","9"]],"card_rows":[["1"],["5","3"],["20","33","9"],["77","225","162","27"]],"row_sums_preview":["1","8","62","491","3975","32730","272940","2298245"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["basis A013610","D+1"],"edges":["e-9081e2b3ba0d880f","e-0b0dcb9e814d6110"],"text":"A001263 \u2192 basis A013610 \u2192 D+1"},"headline":"A001263 \u2192 basis A013610 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      5,       3;\n   3:     20,      33,        9;\n   4:     77,     225,      162,       27;\n   5:    294,    1260,     1638,      702,       81;\n   6:   1122,    6390,    12420,     9720,     2835,      243;\n   7:   4290,   30591,    80190,    95175,    51030,    10935,      729;\n   8:  16445,  141141,   468468,   758835,   623700,   246645,    40824,    2187;\n   9:  63206,  634920,  2558556,  5297292,  5999994,  3667356,  1122660,  148716,   6561;\n  10: 243542, 2804490, 13318344, 33729696, 49223538, 41793570, 19901700, 4881384, 531441, 19683;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-4*n^2-3*n+4+(6*n^2-5*n-11)*x+(3*n+3)*x^2)*P(n,x) = (12*n^3-22*n^2+12*n-2+(27*n^3-48*n^2-2*n+13)*x+(54*n^2-114*n+66)*x^2+(27*n-27)*x^3)*P(n-1,x) - ((6*n^2-5*n+1)*x+(18*n^2-3*n-7)*x^2+(36*n-24)*x^3+18*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^3 - 22*n^2 + 12*n - 2 + (27*n^3 - 48*n^2 - 2*n + 13)*x + (54*n^2 - 114*n + 66)*x^2 + (27*n - 27)*x^3)*P[n - 1, x] - ((6*n^2 - 5*n + 1)*x + (18*n^2 - 3*n - 7)*x^2 + (36*n - 24)*x^3 + 18*x^4)*D[P[n - 1, x], x])/(3*n^3 - 4*n^2 - 3*n + 4 + (6*n^2 - 5*n - 11)*x + (3*n + 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((12*n^3 - 22*n^2 + 12*n - 2 + (27*n^3 - 48*n^2 - 2*n + 13)*x + (54*n^2 - 114*n + 66)*x^2 + (27*n - 27)*x^3)*v[i - 1] - ((6*n^2 - 5*n + 1)*x + (18*n^2 - 3*n - 7)*x^2 + (36*n - 24)*x^3 + 18*x^4)*deriv(v[i - 1], x))/(3*n^3 - 4*n^2 - 3*n + 4 + (6*n^2 - 5*n - 11)*x + (3*n + 3)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A013610","D+1"],"edges":["e-21b441a6a6a148f7","e-f9847bed23834472"],"text":"narayana \u2192 basis A013610 \u2192 D+1"},"headline":"narayana \u2192 basis A013610 \u2192 D+1","chain":null}
