{"id":"n-af0f86ab2b07c466","revision":0,"first_index":0,"fingerprint":"f6-5ee05b1a0fa9fb0d","name":"one_plus_derivative of basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^n.","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":true,"rows":[["1"],["23","15"],["304","690","225"],["3392","13680","15525","3375"],["34816","203520","410400","310500","50625"],["339968","2611200","7632000","10260000","5821875","759375"],["3211264","30597120","117504000","228960000","230850000","104793750","11390625"],["29622272","337182720","1606348800","4112640000","6010200000","4847850000","1833890625","170859375"],["268435456","3554672640","20230963200","64253952000","123379200000","144244800000","96957000000","31438125000","2562890625"],["2399141888","36238786560","239940403200","910393344000","2168570880000","3331238400000","3245508000000","1869885000000","530518359375","38443359375"],["21206401024","359871283200","2717908992000","11997020160000","34139750400000","65057126400000","83280960000000","69546600000000","35060343750000","8841972656250","576650390625"],["185757335552","3499056168960","29689380864000","149484994560000","494877081600000","1126611763200000","1789070976000000","1963051200000000","1434398625000000","642772968750000","145892548828125","8649755859375"],["1614907703296","33436320399360","314915055206400","1781362851840000","6726824755200000","17815574937600000","33798352896000000","46004682240000000","44168652000000000","28687972500000000","11569913437500000","2387332617187500","129746337890625"],["13950053777408","314907002142720","3260041238937600","20469478588416000","86841439027200000","262346165452800000","579006185472000000","941525544960000000","1121364129600000000","956987460000000000","559415463750000000","205103010937500000","38794155029296875","1946195068359375"],["119846767427584","2929511293255680","33065235224985600","228202886725632000","1074647625891840000","3647340439142400000","9182115790848000000","17370185564160000000","24715045555200000000","26165163024000000000","20096736660000000000","10679749762500000000","3589302691406250000","626674812011718750","29192926025390625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","38","1219","35972","1009861","27424418","727306759","18948593792","486890038921","12373135674398","311599410433099","7788184108166012","193420380856527181","4777388126180429978","117440472331196900239"],"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":[["-7","15"],["15","0"]]},"terms":[{"coeff":{"coeffs":[["64","240","225"],["120","225","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191452,"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-ea9e15e1475cb103"]},{"id":191453,"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-ea9e15e1475cb103"]},{"id":191454,"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-ea9e15e1475cb103"]},{"id":191455,"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-ea9e15e1475cb103"]}],"region_exact":{},"incoming_edges":[{"id":"e-ea9e15e1475cb103","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-af0f86ab2b07c466","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":"2b68711bde6615f9f152af36b1321124f2946e9ab3bdb04c1774273e93098dfb","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-a6cd3f3657ee08fd","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^n.","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:          1;\n  1:         23,          15;\n  2:        304,         690,          225;\n  3:       3392,       13680,        15525,         3375;\n  4:      34816,      203520,       410400,       310500,         50625;\n  5:     339968,     2611200,      7632000,     10260000,       5821875,        759375;\n  6:    3211264,    30597120,    117504000,    228960000,     230850000,     104793750,      11390625;\n  7:   29622272,   337182720,   1606348800,   4112640000,    6010200000,    4847850000,    1833890625,     170859375;\n  8:  268435456,  3554672640,  20230963200,  64253952000,  123379200000,  144244800000,   96957000000,   31438125000,   2562890625;\n  9: 2399141888, 36238786560, 239940403200, 910393344000, 2168570880000, 3331238400000, 3245508000000, 1869885000000, 530518359375, 38443359375;\n","recurrence_sentence":"The row polynomials satisfy (15*n-7+15*x)*P(n,x) = (120*n+64+(225*n+240)*x+225*x^2)*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[((120*n + 64 + (225*n + 240)*x + 225*x^2)*P[n - 1, x])/(15*n - 7 + 15*x)]];\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] = ((120*n + 64 + (225*n + 240)*x + 225*x^2)*v[i - 1])/(15*n - 7 + 15*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-c44f7114310405ad","steps":["basis A013610","D+1"],"edges":["e-c78a98c6ca91439c","e-ea9e15e1475cb103"],"text":"A013622 \u2192 basis A013610 \u2192 D+1"},"headline":"A013622 \u2192 basis A013610 \u2192 D+1","chain":null}