{"id":"n-044452246344f122","revision":0,"first_index":0,"fingerprint":"f6-1f944cc0b901d444","name":"one_plus_derivative of basis_map_A013611 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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","4"],["25","52","16"],["98","372","336","64"],["378","2128","3584","1920","256"],["1452","10920","27840","28160","10240","1024"],["5577","52668","182160","283200","195840","52224","4096"],["21450","244244","1073072","2291520","2464000","1254400","258048","16384"],["82654","1102816","5893888","16144128","24088064","19238912","7569408","1245184","65536"],["319124","4884984","30805632","103434240","199600128","223082496","138657792","43646976","5898240","262144"],["1234506","21330920","155195040","618658560","1473239040","2144722944","1878589440","939294720","242810880","27525120","1048576"],["4784276","92122184","760057760","3513206400","9987056640","18047361024","20717469696","14684651520","6055526400","1312030720","126877696","4194304"],["18572500","394390752","3640091136","19160049920","63442368000","137526681600","196990599168","183975542784","108091146240","37486592000","6920601600","578813952","16777216"],["72209880","1676576720","17121170560","101169429504","382920851456","970829223936","1675467915264","1961523412992","1525629321216","757079212032","224319766528","35769024512","2617245696","67108864"],["281150505","7085928780","79339937040","520279614400","2217840584960","6449823091712","13067174055936","18505066364928","18116848189440","11950763016192","5085431070720","1304206245888","181680472064","11744051200","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","93","870","8266","79636","775765","7623118","75430590","750591756","7503649746","75305338620","758242226868","7656262469160","77497832209221"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","-14","8"],["-4","9","4"],["6","8","0"],["4","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-14","8","0"],["4","-32","-20","48"],["16","40","96","0"],["16","48","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-10","-16","-32"],["0","-8","-48","-64","0"],["0","-8","-32","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183075,"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-de3199a8b06f7308"]},{"id":183076,"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-de3199a8b06f7308"]},{"id":183077,"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-de3199a8b06f7308"]},{"id":183078,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-de3199a8b06f7308"]}],"region_exact":{},"incoming_edges":[{"id":"e-de3199a8b06f7308","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-044452246344f122","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":"1cc49d5537daa7d6cc64dd2dda93d185de750c8f532770763aae5badbaf7cb2a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-333a80cbe99b35f1","role":"sequence","name":"basis_map_A013611 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.","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-4036530c3dcad374","kind":"fingerprint_related","node_a_id":"n-044452246344f122","node_b_id":"n-780e1545553d103c","fingerprint":"f6-1f944cc0b901d444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f944cc0b901d444"},"other_id":"n-780e1545553d103c","other":{"id":"n-780e1545553d103c","name":"one_plus_derivative of basis_map_A013611 of narayana","first_index":1,"fingerprint":"f6-1f944cc0b901d444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["25","52","16"]],"card_rows":[["1"],["6","4"],["25","52","16"],["98","372","336","64"]],"row_sums_preview":["1","10","93","870","8266","79636","775765","7623118"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A013611","D+1"],"edges":["e-98249f41ee5a34be","e-acc4b6edfc606490"],"text":"narayana \u2192 basis A013611 \u2192 D+1"},"headline":"narayana \u2192 basis A013611 \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:      1;\n  1:      6,       4;\n  2:     25,      52,       16;\n  3:     98,     372,      336,        64;\n  4:    378,    2128,     3584,      1920,       256;\n  5:   1452,   10920,    27840,     28160,     10240,      1024;\n  6:   5577,   52668,   182160,    283200,    195840,     52224,      4096;\n  7:  21450,  244244,  1073072,   2291520,   2464000,   1254400,    258048,    16384;\n  8:  82654, 1102816,  5893888,  16144128,  24088064,  19238912,   7569408,  1245184,   65536;\n  9: 319124, 4884984, 30805632, 103434240, 199600128, 223082496, 138657792, 43646976, 5898240, 262144;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+6*n^2-4*n+(8*n^2+9*n-14)*x+(4*n+8)*x^2)*P(n,x) = (16*n^3+16*n^2+4*n+(48*n^3+40*n^2-32*n-14)*x+(96*n^2-20*n+8)*x^2+48*n*x^3)*P(n-1,x) - ((8*n^2+8*n+2)*x+(32*n^2+48*n+10)*x^2+(64*n+16)*x^3+32*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 16*n^2 + 4*n + (48*n^3 + 40*n^2 - 32*n - 14)*x + (96*n^2 - 20*n + 8)*x^2 + 48*n*x^3)*P[n - 1, x] - ((8*n^2 + 8*n + 2)*x + (32*n^2 + 48*n + 10)*x^2 + (64*n + 16)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(4*n^3 + 6*n^2 - 4*n + (8*n^2 + 9*n - 14)*x + (4*n + 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 16*n^2 + 4*n + (48*n^3 + 40*n^2 - 32*n - 14)*x + (96*n^2 - 20*n + 8)*x^2 + 48*n*x^3)*v[i - 1] - ((8*n^2 + 8*n + 2)*x + (32*n^2 + 48*n + 10)*x^2 + (64*n + 16)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(4*n^3 + 6*n^2 - 4*n + (8*n^2 + 9*n - 14)*x + (4*n + 8)*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-c2eaf5212a4dec13","steps":["basis A013611","D+1"],"edges":["e-c1ad06aed8ab4413","e-de3199a8b06f7308"],"text":"A001263 \u2192 basis A013611 \u2192 D+1"},"headline":"A001263 \u2192 basis A013611 \u2192 D+1","chain":null}