{"id":"n-b4a0a2fa43623102","revision":0,"first_index":1,"fingerprint":"f6-861ee3c08734918c","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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"],["7","4"],["32","56","16"],["121","388","336","64"],["415","1992","3120","1792","256"],["1344","8688","20256","20864","8960","1024"],["4193","34248","107376","163840","125440","43008","4096"],["12743","126092","499728","1023040","1155840","703488","200704","16384"],["37984","442024","2124848","5494272","8309760","7440384","3756032","917504","65536"],["111545","1493516","8456160","26521216","50604288","60533760","44843008","19333120","4128768","262144"],["323711","4903824","31995168","118243072","273521920","412274688","407314432","257163264","96731136","18350080","1048576"],["930432","15736800","116340288","495783680","1351278080","2467526656","3068190720","2579595264","1418526720","473169920","80740352","4194304"],["2653057","49564560","409678368","1979687680","6222926080","13384544256","20147527680","21296185344","15574302720","7583825920","2272264192","352321536","16777216"],["7514311","153695764","1405018848","7596875392","27087217920","67185957888","119067574272","152246943744","139843731456","90458030080","39516635136","10745806848","1526726656","67108864"],["21161120","470365976","4713159760","28206706176","112579181568","316761384960","647216467968","973836779520","1081302908928","877722599424","508837232640","201527918592","50163875840","6576668672","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","104","909","7575","61136","482201","3738019","28588344","216287525","1621869871","12072013216","89292258609","656908837179","4810204846600"],"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":[["-248","-192","128"],["504","192","-128"],["-384","0","0"],["128","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["100","144","-64","-256"],["0","0","0","0"],["192","256","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["35","128","-160","-512","-256"],["-80","-384","-256","0","0"],["80","384","256","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181973,"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-544b8840ca26c84c"]},{"id":181974,"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-544b8840ca26c84c"]},{"id":181975,"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-544b8840ca26c84c"]},{"id":181976,"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-544b8840ca26c84c"]}],"region_exact":{},"incoming_edges":[{"id":"e-544b8840ca26c84c","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-b4a0a2fa43623102","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":"3d449f292263604b4b57ff98d029fa8e8052665838433872474a4eb576665561","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-02f1a3354aa99430","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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":[],"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:      7,       4;\n   3:     32,      56,      16;\n   4:    121,     388,     336,       64;\n   5:    415,    1992,    3120,     1792,      256;\n   6:   1344,    8688,   20256,    20864,     8960,     1024;\n   7:   4193,   34248,  107376,   163840,   125440,    43008,     4096;\n   8:  12743,  126092,  499728,  1023040,  1155840,   703488,   200704,    16384;\n   9:  37984,  442024, 2124848,  5494272,  8309760,  7440384,  3756032,   917504,   65536;\n  10: 111545, 1493516, 8456160, 26521216, 50604288, 60533760, 44843008, 19333120, 4128768, 262144;\n","recurrence_sentence":"The row polynomials satisfy (128*n^3-384*n^2+504*n-248+(192*n-192)*x-(128*n-128)*x^2)*P(n,x) = (192*n^3+100*n+(256*n^3+144*n)*x-64*n*x^2-256*n*x^3)*P(n-1,x) + (80*n^2-80*n+35+(384*n^2-384*n+128)*x+(256*n^2-256*n-160)*x^2-512*x^3-256*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[((192*n^3 + 100*n + (256*n^3 + 144*n)*x - 64*n*x^2 - 256*n*x^3)*P[n - 1, x] + (80*n^2 - 80*n + 35 + (384*n^2 - 384*n + 128)*x + (256*n^2 - 256*n - 160)*x^2 - 512*x^3 - 256*x^4)*D[P[n - 1, x], x])/(128*n^3 - 384*n^2 + 504*n - 248 + (192*n - 192)*x - (128*n - 128)*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] = ((192*n^3 + 100*n + (256*n^3 + 144*n)*x - 64*n*x^2 - 256*n*x^3)*v[i - 1] + (80*n^2 - 80*n + 35 + (384*n^2 - 384*n + 128)*x + (256*n^2 - 256*n - 160)*x^2 - 512*x^3 - 256*x^4)*deriv(v[i - 1], x))/(128*n^3 - 384*n^2 + 504*n - 248 + (192*n - 192)*x - (128*n - 128)*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-976cdb6e4a693ec8","steps":["basis A013611","D+1"],"edges":["e-60bc1ac61eca3352","e-544b8840ca26c84c"],"text":"A078812 \u2192 basis A013611 \u2192 D+1"},"headline":"A078812 \u2192 basis A013611 \u2192 D+1","chain":null}