{"id":"n-01512c1ccffbc04e","revision":0,"first_index":0,"fingerprint":"f6-56207361c0b00b39","name":"T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= 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":false,"rows":[["1"],["0","1"],["0","1","6"],["0","1","30","60"],["0","1","126","840","840"],["0","1","510","8820","25200","15120"],["0","1","2046","84480","526680","831600","332640"],["0","1","8190","780780","9609600","30270240","30270240","8648640"],["0","1","32766","7108920","164684520","929728800","1755673920","1210809600","259459200"],["0","1","131070","64307940","2734477200","26207541360","83658294720","104977192320","52929676800","8821612800"],["0","1","524286","580082160","44651946360","704409123600","3588264519840","7318998086400","6536815084800","2514159648000","335221286400"],["0","1","2097150","5225982300","722552292000","18413963124480","144674523433440","451925783749440","637926108019200","425898644371200","129060195264000","14079294028800"],["0","1","8388606","47054812200","11634000424200","473355019368000","5613390032342400","25905901012992000","54385044425712000","56187277866720000","29090168012505600","7124122778572800","647647525324800"],["0","1","33554430","423577195860","186802774158000","12043287491835600","212495851590422400","1415337290477510400","4257819873635328000","6400261017678528000","5044133360185920000","2083805912732544000","420970891461120000","32382376266240000"],["0","1","134217726","3812530307040","2994774467270040","304444637230734000","7914802982075589600","74876419374748992000","314960590626986304000","663187018135561920000","747623254690376064000","463994116568446464000","156474880356098304000","26521166162050560000","1748648318376960000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","7","91","1807","49651","1777447","79587691","4327497727","279393234211","21043100301847","1822706292362011","179438587737162607","19879437087904356691","2457608340439083686407"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A304334","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4"],["0","1","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"],["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":170039,"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-8a5f560a82870648"]},{"id":170040,"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-8a5f560a82870648"]},{"id":170041,"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-8a5f560a82870648"]}],"region_exact":{},"incoming_edges":[{"id":"e-8a5f560a82870648","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-01512c1ccffbc04e","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = (A + B D + C D^2 with A = -2 + 2 n, B = -1 + n - 4 x + 4 n x, C = -x + n x) / (n - 1): Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"574bd7b97747d2effab67c27a079dc88bf7bb31366e3fe3a8670954d56c36293","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-68ac132d04cd33c2","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-5af4ea1ce3f85b64","conclusion_name":"derivative of T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n."}],"used_by_total":1,"links":[{"id":"l-537a379cdbb3201c","kind":"oeis_match","node_a_id":"n-01512c1ccffbc04e","node_b_id":null,"fingerprint":null,"oeis_id":"A304334","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A304334","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A304334","relation":"identification"},{"oeis_id":"A304334","relation":"match_triangle"}],"bfiles":[{"node_id":"n-01512c1ccffbc04e","oeis_id":"A304334","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":51537}],"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: 0, 1;\n  2: 0, 1,      6;\n  3: 0, 1,     30,       60;\n  4: 0, 1,    126,      840,        840;\n  5: 0, 1,    510,     8820,      25200,       15120;\n  6: 0, 1,   2046,    84480,     526680,      831600,      332640;\n  7: 0, 1,   8190,   780780,    9609600,    30270240,    30270240,      8648640;\n  8: 0, 1,  32766,  7108920,  164684520,   929728800,  1755673920,   1210809600,   259459200;\n  9: 0, 1, 131070, 64307940, 2734477200, 26207541360, 83658294720, 104977192320, 52929676800, 8821612800;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2)*x*P(n-1,x) + ((n-1)*x+(4*n-4)*x^2)*d/dx P(n-1,x) + (n-1)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2)*x*P[n - 1, x] + ((n - 1)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x] + (n - 1)*x^2*D[P[n - 1, x], {x, 2}])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2)*x*v[i - 1] + ((n - 1)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x) + (n - 1)*x^2*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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}},{"oeis_id":"A304334","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 1,      6;\n  3: 0, 1,     30,       60;\n  4: 0, 1,    126,      840,        840;\n  5: 0, 1,    510,     8820,      25200,       15120;\n  6: 0, 1,   2046,    84480,     526680,      831600,      332640;\n  7: 0, 1,   8190,   780780,    9609600,    30270240,    30270240,      8648640;\n  8: 0, 1,  32766,  7108920,  164684520,   929728800,  1755673920,   1210809600,   259459200;\n  9: 0, 1, 131070, 64307940, 2734477200, 26207541360, 83658294720, 104977192320, 52929676800, 8821612800;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2)*x*P(n-1,x) + ((n-1)*x+(4*n-4)*x^2)*d/dx P(n-1,x) + (n-1)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2)*x*P[n - 1, x] + ((n - 1)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x] + (n - 1)*x^2*D[P[n - 1, x], {x, 2}])/(n - 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2)*x*v[i - 1] + ((n - 1)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x) + (n - 1)*x^2*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-01512c1ccffbc04e","steps":[],"edges":[],"text":"A304334"},"headline":"A304334: T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.","chain":null}