{"id":"n-310c5e0cb618ceb4","revision":0,"first_index":0,"fingerprint":"f6-f1305d7953ba0815","name":"one_plus_derivative of reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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"],["2","1"],["4","5","1"],["7","16","8","1"],["11","40","36","11","1"],["16","85","119","64","14","1"],["22","161","322","264","100","17","1"],["29","280","756","870","495","144","20","1"],["37","456","1596","2442","1925","832","196","23","1"],["46","705","3102","6072","6292","3731","1295","256","26","1"],["56","1045","5643","13728","18018","13923","6580","1904","324","29","1"],["67","1496","9724","28743","46475","45136","27608","10812","2679","400","32","1"],["79","2080","16016","56485","110110","130832","99960","50388","16815","3640","484","35","1"],["92","2821","25389","105248","243100","346086","321708","201552","86184","25025","4807","576","38","1"],["106","3745","38948","187408","505648","848198","939930","713184","377454","139909","35926","6200","676","41","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","10","32","99","299","887","2595","7508","21526","61251","173173","486925","1362627","3797374"],"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":[["-2","2","1"],["5","-4","-2"],["-3","0","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["-2","2","1","0"],["3","-6","-5","-1"],["1","1","0","0"],["2","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-8","-18","-8","-1"],["0","0","0","0","0"],["0","4","1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172437,"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-deafdbdc16ef544b"]},{"id":172438,"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-deafdbdc16ef544b"]},{"id":172439,"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-deafdbdc16ef544b"]}],"region_exact":{},"incoming_edges":[{"id":"e-deafdbdc16ef544b","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-310c5e0cb618ceb4","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":"514b7f76116e401d2dce29e1b39175866ce950ef9c4ef0a0e6109f182e6bf587","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4c232e962100ae75","role":"sequence","name":"reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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-422e2da5dce86d4b","kind":"fingerprint_related","node_a_id":"n-e420707ba29fb085","node_b_id":"n-310c5e0cb618ceb4","fingerprint":"f6-f1305d7953ba0815","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f1305d7953ba0815"},"other_id":"n-e420707ba29fb085","other":{"id":"n-e420707ba29fb085","name":"one_plus_derivative of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-f1305d7953ba0815","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["4","5","1"]],"row_sums_preview":["1","1","3","10","32","99","299","887"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01fe968f3e05a927","steps":["D+1"],"edges":["e-05447d659c882bb2"],"text":"A165253 \u2192 D+1"},"headline":"A165253 \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:  2,   1;\n  2:  4,   5,    1;\n  3:  7,  16,    8,    1;\n  4: 11,  40,   36,   11,    1;\n  5: 16,  85,  119,   64,   14,    1;\n  6: 22, 161,  322,  264,  100,   17,    1;\n  7: 29, 280,  756,  870,  495,  144,   20,   1;\n  8: 37, 456, 1596, 2442, 1925,  832,  196,  23,  1;\n  9: 46, 705, 3102, 6072, 6292, 3731, 1295, 256, 26, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-3*n^2+5*n-2-(4*n-2)*x-(2*n-1)*x^2)*P(n,x) = (2*n^3+n^2+3*n-2+(n^3+n^2-6*n+2)*x-(5*n-1)*x^2-n*x^3)*P(n-1,x) + ((4*n^2-8)*x+(n^2-18)*x^2-8*x^3-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[((2*n^3 + n^2 + 3*n - 2 + (n^3 + n^2 - 6*n + 2)*x - (5*n - 1)*x^2 - n*x^3)*P[n - 1, x] + ((4*n^2 - 8)*x + (n^2 - 18)*x^2 - 8*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + 5*n - 2 - (4*n - 2)*x - (2*n - 1)*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] = ((2*n^3 + n^2 + 3*n - 2 + (n^3 + n^2 - 6*n + 2)*x - (5*n - 1)*x^2 - n*x^3)*v[i - 1] + ((4*n^2 - 8)*x + (n^2 - 18)*x^2 - 8*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + 5*n - 2 - (4*n - 2)*x - (2*n - 1)*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-6758fe7729845be8","steps":["reverse","D+1"],"edges":["e-597fd348d5cdfa65","e-deafdbdc16ef544b"],"text":"A054142 \u2192 reverse \u2192 D+1"},"headline":"A054142 \u2192 reverse \u2192 D+1","chain":null}
