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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":"0d84d52ee1f5baac909488745a68d5ba97a64e696fa06e8bf6e12151fbc72e4a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-37cbe2545ca04836","kind":"oeis_match","node_a_id":"n-735e311cbe9a718e","node_b_id":null,"fingerprint":null,"oeis_id":"A134980","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A134980","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-48b7b4c2eb8fba0c","kind":"oeis_match","node_a_id":"n-735e311cbe9a718e","node_b_id":null,"fingerprint":null,"oeis_id":"A397048","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A397048","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A397048","relation":"identification"},{"oeis_id":"A134980","relation":"match_row_sums"},{"oeis_id":"A397048","relation":"match_triangle"}],"bfiles":[{"node_id":"n-735e311cbe9a718e","oeis_id":"A134980","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":42574},{"node_id":"n-735e311cbe9a718e","oeis_id":"A397048","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":32746}],"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,         1;\n  2:         4,         5,         1;\n  3:        27,        37,        12,         1;\n  4:       256,       369,       151,        22,        1;\n  5:      3125,      4651,      2190,       425,       35,       1;\n  6:     46656,     70993,     36751,      8550,      965,      51,      1;\n  7:    823543,   1273609,    706104,    186501,    25830,    1904,     70,    1;\n  8:  16777216,  26269505,  15341887,   4453638,   714021,   65562,   3402,   92,   1;\n  9: 387420489, 612579511, 372684090, 116419465, 20853525, 2245467, 146664, 5646, 117, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) + (n+2*x)*d/dx P(n-1,x) + x*d^2/dx^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[(n + x)*P[n - 1, x] + (n + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {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] = (n + x)*v[i - 1] + (n + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), 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}},{"oeis_id":"A397048","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:         1,         1;\n  2:         4,         5,         1;\n  3:        27,        37,        12,         1;\n  4:       256,       369,       151,        22,        1;\n  5:      3125,      4651,      2190,       425,       35,       1;\n  6:     46656,     70993,     36751,      8550,      965,      51,      1;\n  7:    823543,   1273609,    706104,    186501,    25830,    1904,     70,    1;\n  8:  16777216,  26269505,  15341887,   4453638,   714021,   65562,   3402,   92,   1;\n  9: 387420489, 612579511, 372684090, 116419465, 20853525, 2245467, 146664, 5646, 117, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) + (n+2*x)*d/dx P(n-1,x) + x*d^2/dx^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[(n + x)*P[n - 1, x] + (n + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (n + x)*v[i - 1] + (n + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); 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-735e311cbe9a718e","steps":[],"edges":[],"text":"A397048"},"headline":"A397048: Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.","chain":null}