{"id":"n-c9c651476acf61cd","revision":0,"first_index":1,"fingerprint":"f6-2513bba9c35d0db8","name":"one_plus_derivative of gamma_polynomial of narayana","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","1"],["4","3"],["7","10","2"],["11","30","10"],["16","75","45","5"],["22","161","175","35"],["29","308","560","196","14"],["37","540","1512","924","126"],["46","885","3570","3570","840","42"],["56","1375","7590","11550","4620","462"],["67","2046","14850","32340","20790","3564","132"],["79","2938","27170","80652","78078","22308","1716"],["92","4095","47047","183183","252252","114114","15015","429"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","3","7","19","51","141","393","1107","3139","8953","25653","73789","212941","616227"],"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":[["8","-15","4"],["3","-11","4"],["-4","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["4","-55","92","-32"],["1","49","-72","16"],["-2","-24","16","0"],["1","4","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-38","64","-32"],["0","-6","32","-32","0"],["0","2","-8","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172176,"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-b2f59e7a5fd1613e"]},{"id":172177,"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-b2f59e7a5fd1613e"]},{"id":172178,"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-b2f59e7a5fd1613e"]}],"region_exact":{},"incoming_edges":[{"id":"e-b2f59e7a5fd1613e","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-c9c651476acf61cd","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":"d13fb0116bba793ded0e26065d756762e2be1d540b8d1e13f78cb2ff75d835fe","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6bb2422ce01062d8","role":"sequence","name":"gamma_polynomial of narayana","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":[{"id":"l-942954e64c4281ed","kind":"fingerprint_related","node_a_id":"n-9f412b3a63f0fc43","node_b_id":"n-c9c651476acf61cd","fingerprint":"f6-2513bba9c35d0db8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2513bba9c35d0db8"},"other_id":"n-9f412b3a63f0fc43","other":{"id":"n-9f412b3a63f0fc43","name":"one_plus_derivative of Triangular array of Motzkin polynomial coefficients.","first_index":0,"fingerprint":"f6-2513bba9c35d0db8","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["4","3"]],"row_sums_preview":["1","1","3","7","19","51","141","393"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-169bd483754ab390","steps":["D+1"],"edges":["e-a191b7e0aef9387b"],"text":"A055151 \u2192 D+1"},"headline":"A055151 \u2192 D+1","chain":null}}],"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:  1;\n   3:  2,   1;\n   4:  4,   3;\n   5:  7,  10,    2;\n   6: 11,  30,   10;\n   7: 16,  75,   45,   5;\n   8: 22, 161,  175,  35;\n   9: 29, 308,  560, 196,  14;\n  10: 37, 540, 1512, 924, 126;\n","recurrence_sentence":"The row polynomials satisfy (n^3-4*n^2+3*n+8+(4*n^2-11*n-15)*x+(4*n+4)*x^2)*P(n,x) = (n^3-2*n^2+n+4+(4*n^3-24*n^2+49*n-55)*x+(16*n^2-72*n+92)*x^2+(16*n-32)*x^3)*P(n-1,x) + ((2*n^2-6*n+6)*x-(8*n^2-32*n+38)*x^2-(32*n-64)*x^3-32*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[((n^3 - 2*n^2 + n + 4 + (4*n^3 - 24*n^2 + 49*n - 55)*x + (16*n^2 - 72*n + 92)*x^2 + (16*n - 32)*x^3)*P[n - 1, x] + ((2*n^2 - 6*n + 6)*x - (8*n^2 - 32*n + 38)*x^2 - (32*n - 64)*x^3 - 32*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 3*n + 8 + (4*n^2 - 11*n - 15)*x + (4*n + 4)*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] = ((n^3 - 2*n^2 + n + 4 + (4*n^3 - 24*n^2 + 49*n - 55)*x + (16*n^2 - 72*n + 92)*x^2 + (16*n - 32)*x^3)*v[i - 1] + ((2*n^2 - 6*n + 6)*x - (8*n^2 - 32*n + 38)*x^2 - (32*n - 64)*x^3 - 32*x^4)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 3*n + 8 + (4*n^2 - 11*n - 15)*x + (4*n + 4)*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-2a05d8340c329b21","steps":["\u03b3","D+1"],"edges":["e-dbb6454df235fcfe","e-b2f59e7a5fd1613e"],"text":"narayana \u2192 \u03b3 \u2192 D+1"},"headline":"narayana \u2192 \u03b3 \u2192 D+1","chain":null}