{"id":"n-414d59e057d30255","revision":0,"first_index":0,"fingerprint":"f6-3a1ad01786490d10","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["11","8"],["58","144","48"],["275","1200","1248","256"],["1246","7840","14880","8960","1280"],["5502","45360","127680","143360","57600","6144"],["23892","243936","917280","1559040","1182720","344064","28672"],["102531","1249248","5898816","13601280","15859200","8773632","1949696","131072"],["436150","6177600","35171136","103014912","164989440","141926400","60211200","10616832","589824"],["1842698","29755440","198506880","707266560","1452971520","1734340608","1154334720","389283840","56033280","2621440"],["7742956","140414560","1074696480","4518159360","11377766400","17564811264","16359383040","8715632640","2400583680","288358400","11534336"],["32391086","651819168","5631362880","27314327040","81662929920","155574042624","188735619072","141699317760","61993451520","14244904960","1453326336","50331648"],["134999788","2985388224","28742121408","158092014080","548060198400","1246384668672","1876925546496","1846814441472","1145402818560","419849830400","81870716928","7197425664","218103808"],["560889500","13520780000","143558233728","883328782336","3487129085440","9237208780800","16686570700800","20487022313472","16741774393344","8743372718080","2729023897600","458017996800","35114713088","939524096"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","19","250","2979","34206","385646","4299604","47565475","523133494","5726956986","62459083116","679043824014","7362678273900","79647142809084"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-5","15","-4"],["15","-23","4"],["-14","8","0"],["4","0","0"]]},"terms":[{"coeff":{"coeffs":[["-6","-46","-80","32"],["-2","66","260","-80"],["40","68","-276","48"],["-48","-136","96","0"],["16","48","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-15","-24","16"],["0","-2","14","80","-32"],["0","12","32","-64","0"],["0","-8","-32","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183358,"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-49c95d9338f284fb"]},{"id":183359,"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-49c95d9338f284fb"]},{"id":183360,"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-49c95d9338f284fb"]},{"id":183361,"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-49c95d9338f284fb"]}],"region_exact":{},"incoming_edges":[{"id":"e-49c95d9338f284fb","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-414d59e057d30255","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":"adefee7acd04e98d082537682e19127282b16729c90f62e3d6fb515361cb6dbe","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b8e11e765570cb44","role":"sequence","name":"basis_map_A013611 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","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:      1;\n  2:     11,       8;\n  3:     58,     144,       48;\n  4:    275,    1200,     1248,       256;\n  5:   1246,    7840,    14880,      8960,      1280;\n  6:   5502,   45360,   127680,    143360,     57600,      6144;\n  7:  23892,  243936,   917280,   1559040,   1182720,    344064,    28672;\n  8: 102531, 1249248,  5898816,  13601280,  15859200,   8773632,  1949696,   131072;\n  9: 436150, 6177600, 35171136, 103014912, 164989440, 141926400, 60211200, 10616832, 589824;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-14*n^3+15*n^2-5*n+(8*n^3-23*n^2+15*n)*x+(4*n^2-4*n)*x^2)*P(n,x) = (16*n^4-48*n^3+40*n^2-2*n-6+(48*n^4-136*n^3+68*n^2+66*n-46)*x+(96*n^3-276*n^2+260*n-80)*x^2+(48*n^2-80*n+32)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+2*n+2)*x+(32*n^3-32*n^2-14*n+15)*x^2+(64*n^2-80*n+24)*x^3+(32*n-16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 48*n^3 + 40*n^2 - 2*n - 6 + (48*n^4 - 136*n^3 + 68*n^2 + 66*n - 46)*x + (96*n^3 - 276*n^2 + 260*n - 80)*x^2 + (48*n^2 - 80*n + 32)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 2*n + 2)*x + (32*n^3 - 32*n^2 - 14*n + 15)*x^2 + (64*n^2 - 80*n + 24)*x^3 + (32*n - 16)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 14*n^3 + 15*n^2 - 5*n + (8*n^3 - 23*n^2 + 15*n)*x + (4*n^2 - 4*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 48*n^3 + 40*n^2 - 2*n - 6 + (48*n^4 - 136*n^3 + 68*n^2 + 66*n - 46)*x + (96*n^3 - 276*n^2 + 260*n - 80)*x^2 + (48*n^2 - 80*n + 32)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 2*n + 2)*x + (32*n^3 - 32*n^2 - 14*n + 15)*x^2 + (64*n^2 - 80*n + 24)*x^3 + (32*n - 16)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 14*n^3 + 15*n^2 - 5*n + (8*n^3 - 23*n^2 + 15*n)*x + (4*n^2 - 4*n)*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-b1b19325e0094118","steps":["basis A013611","D+1"],"edges":["e-a70bef035c61e98a","e-49c95d9338f284fb"],"text":"A367270 \u2192 basis A013611 \u2192 D+1"},"headline":"A367270 \u2192 basis A013611 \u2192 D+1","chain":null}